One of our drivers spotted him first – a stranger in fatigues who had slipped into the open-air restaurant by the Tigris River in central Baghdad and was watching us intently. Two other unknown men lurked in the background. It was dark, we had just finished three large dishes of masgouf – roasted carp – and were sitting around a blazing fire.

The driver took no chances. He came over and whispered in my ear: “Mr Martin, we must go.” We were out of the restaurant in a trice, walking hastily to our two cars, our bodyguards with their pistols drawn. Within another minute we were speeding down Abu Niwas road towards the safety of our heavily fortified hotel.

Necessary or not, our rapid departure brought a sudden and sobering end to our day out in Baghdad – an attempt to test whether, given the recent drop in violence, it is once again possible for a Westerner to venture out into one of the world’s most dangerous cities. Can he shop, visit a barber, eat a meal or lead anything like a normal life without risking kidnap and murder? Can he experience more of Baghdad than jittery car rides to the airport and the Green Zone or patrols in US humvees?

We had sat down with our security contractors and planned the day with military precision – locations, routes, timetables. We confined ourselves to central Baghdad, the outer areas still being far too dangerous. Even here certain locations were rejected as too risky. The zoo, for example, was thought to be a hang-out of the Mahdi army, the Shia militia.

We agreed ground rules: no more than 20 minutes in any one place, cars outside and running at all times, no dawdling on pavements, no hanging around if crowds gathered, text messages to confirm our arrival and departure from each location. One bodyguard would stand outside while the other came with me. They would carry discreet handguns, not ostentatious rifles. Above all, though, I would be relying on the quick wits and sixth sense of my Iraqi team.

The seven of us – the drivers, bodyguards, a fixer, an interpreter and I – left the hotel at 8.30am on Sunday, the first day of the working week.

The streets were still fairly empty, Baghdad not being a city that opens early. Ten minutes later I walked nervously into the al-Warda supermarket in the relatively affluent district of Karrada. The manager, Bassim Oubayis, 33, stopped spooning honey onto his flat Iraqi bread and grinned in surprise. After the US invasion of 2003 he used to get 40 or 50 Western customers a day, he said, but for two years he had seen hardly any.

The shop had Energiser batteries, Gillette razors, Wrigley’s chewing gum – anything that you could want except ham and bacon which Oubayis removed from the shelves two years ago for fear of Islamic zealots. I bought soup, butter, chocolate and apple juice. It was the first time I had used Iraqi dinars, not US dollars, since 2003.

“Business is getting better,” said Oubayis, offering me a mango juice. He had recently dispensed with the four men who had guarded his shops with Kalashnikovs. A few customers had drifted back from self-imposed exile in Jordan and Syria. His staff were no longer arriving late because of bombs or shootings. He was staying open an extra two or three hours in the evenings. “Having you here reflects the improved security. Under no circumstances could you have come before,” he said.

I asked if he had lost relatives in the fighting. Just a cousin killed for joining the Iraqi Army, he replied. An assistant said that he had lost two relatives to a car bomb. Another shopworker had had a cousin killed by an improvised explosive device as she crossed a road. They spoke of their losses as if they were entirely normal – which in Baghdad they are.

Outside, I switched cars as a precaution and we headed east up Arassat Street – a curious cross between a war-zone and flourishing commercial strip. Side roads are barricaded with rocks and tree trunks. There are armed police at every junction. VIP motor-cades weave through the traffic with lights flashing and sirens wailing. The more important buildings are protected by concrete blast barriers and coiled razor wire. At the same time street cleaners are sweeping away rubbish, gardeners tend newly-planted shrubs and grass in the central reservations, and great banks of fridges, freezers, satellite dishes, televisions and generators cover the pavements outside electrical shops.

We stop at B-town, a modern mini-mall selling everything from Perrier water and Umbro trainers to hookahs, but not the scarf I need (nights here are freezing). The manager’s face lights up as I enter. Jalil Khalid, 39, says I’m his first Western customer since last summer. “It makes me happy. It makes me think things are getting better,” he remarks. But, he adds, “every time there’s any stability and people start coming back another bomb goes off and they vanish again”. Khalid certainly has not had much to celebrate. B-town’s owner opened the store straight after the US invasion of 2003, hoping for an economic boom, but fled to Germany as Iraq descended into mayhem. Its wealthiest customers decamped to Jordan, Egypt or Dubai.

A year ago a suicide bomber targeted a religious march as it passed outside and blew in all the windows. An employee was killed in an explosion late last year as he walked home. Khalid had lost no immediate family, largely because they never go outside, but four cousins had died in a Baghdad market bombing. As for friends, so many had been killed, injured or gone abroad that “we have chiselled the word ‘friend’ from the dictionary”.

The drivers constantly check that we are not being followed as we inch up congested Sadoun Street. We pass Saddam Hussain’s old Air Force headquarters, now looted and occupied by squatters, the bombed-out telephone exchange, the concrete shell of the former Ministry of Youth. But this is a city ravaged not just by the strife of the last five years. It has suffered from more than two decades of nonstop war, sanctions and deprivation.

The streets are rutted, the buildings crumbling and dilapidated, the lampposts broken. Walls are pockmarked by shrapnel. Everything is covered in a fine brown dust. Someone somewhere is making an effort, however. Many of the ugly concrete blast barriers have recently been decorated with brightly coloured murals of bucolic scenes. Where Saddam’s statue once stood in Fardous Square there is now a children’s playground – the authorities’ antidote to horror. A ring of palm trees has been planted in Tahrir Square. We even spot a street-cleaning lorry – an unprecedented sight in Baghdad.

We turn west onto the al-Jumhuriya bridge over the silted-up Tigris and sit in a monstrous traffic jam for 15 minutes, waiting to pass through an Iraqi army checkpoint. The cars around us are mostly jalopies – imported fourth-hand from Germany, Japan and Saudi Arabia. Many are self-styled taxis – jobs being so scarce. Destitute women in black hawk boxes of tissues, biscuits, black bin liners.

On the far side we reach a shabby district called Karradat Mariem. We spurn one barber’s shop because it looks too full. We pick an emptier one with a large, faded poster of London’s Tower Bridge on the wall. I go in and ask for a shave. Every Iraqi has a story to tell. My bemused barber, Ahmed Abdullah, 30, is no exception. In the wake of the US invasion he had cut hair on a US military base until insurgents threatened to kill him. Eighteen months ago he had abandoned his shop in smarter Palestine Street after Sunni extremists had killed half-a-dozen of his fellow barbers for daring to shave beards. Even here his windows had recently been blown out by a car bomb.

Did he think the worst of the violence was now over? “Inshallah [God willing],” he replied, as he took out a cutthroat razor to trim my sideburns. I gulped. “Not near my neck,” I joked. He laughed. When he had finished he refused payment.

We were beginning to relax. Near by we noticed a freshly painted restaurant called al-Yasmeen with bright red plastic chairs and tables on the pavement. We ordered glasses of sweet, dark Iraqi tea.

Abbas Fadhill, 30, the manager, told us it had been open only ten days. The previous restaurant on these premises had been bombed six months earlier for serving Iraqi policemen, killing the owner’s brother, son and three customers. “We saw the situation was improving and decided to take a risk,” Fadhill said. “But it’s difficult to get customers to come back because this restaurant has a reputation for bombs.” We drove on past Baghdad’s defunct railway, past old men with typewriters filling out supplicants’ forms outside a court, past the most surprising sight of all – a brand new set of working traffic lights (disobeyed by all). We pulled to the side for a convoy of American humvees – a rare sight now on Baghdad’s streets.

I wanted to buy beer, but the two or three shops still brave enough to sell alcohol were, alas, closed for the Shia religious festival of Ashura. We went instead to Haider Double, Baghdad’s most famous falafel shop. Inside the narrow, white-tiled premises a middle-aged manager named Walid Abdullah wistfully recalled how the shop used to open round the clock and sold 3,000 filled breads each day. Today it closes at 3pm and sells barely 300.

It was too near the Green Zone, he said. There were too many checkpoints, too many parking restrictions, too much tension. “Business is improving, but not significantly.” Around mid-afternoon we arrive at St George’s Anglican church near the bottom of Haifa Street for the Sunday afternoon service. The squat 1930s building is almost entirely obscured by huge blast walls and coiled razor wire. It is flanked on one side by the burnt-out Rasheed theatre, and on the other by the forlorn shell of Saddam’s old Ministry of Information.

The church has had many parishioners killed, including 11 elders seized and murdered in a single incident in 2004. The only item on its last parish council meeting was how to protect Faiz Georges, its lay pastor, from being kidnapped like his two predecessors. A congregation of more than 1,000 adults has shrunk to 250 in four years, and one woman told me this was only the second time she had left her house since Christmas – the other being to visit her parents.

Guards usher me nervously inside where I find a surprise. The church is packed with children, scores of them – another extraordinarily rare sight. “This is the only place they can come and find love and people who care about them,” Georges explains.

Canon Andrew White, the British vicar of Baghdad, arrives from the Green Zone in a big black Ford Suburban with darkened windows, flashing lights and an escort of half a dozen Iraqi police and army vehicles bristling with guns. A great bear of a man, he swaps his flak jacket for his robes and greets his parishioners with hugs and kisses. “It’s not like arriving at a church in England,” he jokes.

The church, replete with a memorial to the British empire’s war dead, is full, mostly with women and children. Many of the men have been killed or kidnapped, Canon White says. “You are serving Jesus in the most dangerous and difficult place in the world,” he tells his Iraqi congregation, and then announces that there is to be a baptism.

In the font at the back of the church, amid a crowd of excited children shouting “Hallelujah” like a football chant, he baptises a baby girl named Raneem. It is an unexpectedly moving ceremony – a celebration of a new life in a city ravaged by death.

By the time we leave it is nearly dusk. We have been welcomed everywhere. We have not felt threatened once. In fact, we have become a touch complacent. We cross back to the east bank of the Tigris and take a walk in the fading light in a newly restored riverside park. There are Baghdadis strolling along by the water or sitting on the benches, young men playing football, kids enjoying themselves on a ridiculously large number of brand new swings and slides. It is a tranquil scene. As the sun sets, it is possible to believe that peace is returning to this tortured city.

Alas, the illusion does not last. We move on to the Faris fish restaurant at the far end of the park for dinner. Until recently it opened only at lunchtimes, but now stays open late into the evening. We have just finished eating, and the manager is recalling how Saddam and his sons used to order masgouf from the restaurant, when the driver whispers urgently in my ear. We have, it seems, attracted too much attention to ourselves. One miscalculation and Baghdad can still be deadly.

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