"Al-Qa'ida killed two of our men here two days ago," the cop said. "Then they called us up to tell us the name of their operation – on a police radio!" We were standing in rebuilt Fallujah, where the police request all foreigners to call by for an escort. We got six, one wearing a ski mask. You get the idea. As a police colonel said later: "Al-Qa'ida [is] still here, they are a nuisance, to me personally when I have to move around the city. But they are not what they were."

We were standing in the old US Marine base not far from the newly re-built railway station – there are, of course, no trains – and the pale stencil of "USMC" was still on the wall. But there was dust blowing around the yard and some of the sandbags had broken open.

All the way back to Baghdad, the old American bases looked scruffy, some of the concrete blast walls had collapsed. There was a feeling of an empire departed – Britain after the Romans had left.

Not that Iraq doesn't have problems. Its vice-president, Tareq al-Hashemi, has fled Baghdad for Iraqi Kurdistan, then flown to Qatar and then to Saudi Arabia and is touring the Gulf – to the rage of Prime Minister Nouri al-Maliki's government, which has charged him with running death squads. Al-Hashemi claims that three of his men have been tortured to death in a Baghdad prison – the al-Maliki administration says only one has died, and for medical reasons.

It's not a good story. The central government even wants to discipline the authorities at the airport at Irbil, the Kurdish capital, for allowing al-Hashemi to fly to Qatar. Britain is not a republic, but it's a bit like Cameron calling Clegg a traitor, Clegg flees to Scotland and Cameron claims Edinburgh airport should not have let him fly to Ireland. "Iraqi law is one and it applies to all, including Kurdistan," a government spokesman announced.

But it doesn't. Iraqi Kurdistan is almost a separate state – it has its own flag, its own language – though at least its car registration plates are still Iraqi. All the way up the highway to Irbil, there are Iraqi police and army checkpoints – some of the police drive around in bright green-and-yellow Chevrolets – and they are a lot friendlier than the old American checkpoints where nervous, frightened US soldiers pointed their rifles at you in case you were a suicider. There's a 40-mile stretch which is generally regarded as unsafe – it's al-Qa'ida territory, insofar as you can use the phrase – and then you're into Kurdistan, and the rivers froth beneath the road and the cops are even friendlier. Outside Irbil, there's a big clearance station that looks like a customs shed. Yes, Iraq is safer, even though the old Sunni "resistance" – without any more American enemies to attack – has now announced its enemy is Iran. Well maybe.

I am on the highway with Dr Lubna Naji, a 25 year-old medical practitioner. She shakes her head. "There is no real country anymore. I talk to my friends, mostly doctors – and all talk of moving out of Iraq. They all dream of going outside. Because home is where you belong – where you are wanted. We've lost our sense of something that belongs to us, our homeland. We've lost our national identity as Iraqis."

You hear this a lot. The government, they tell you in Baghdad, is unashamedly sectarian. And corrupt. Saad Tahr Hussein is a journalist with a sense of purpose. "For me, I prefer to die here, like a tree standing up, not to melt away." I laugh at his mixed metaphor and he waves his hand. "That's the difference between the older and the younger generations." We drive past so many checkpoints together that I'm almost giddy. By the time I'm in Irbil, I've counted 13 different types of camouflage uniform. Those in black around Baghdad are al-Maliki's Shia al-Dawa party. All the cops are militiamen, I'm told, in the town of al-Hawaya. Well, let them all obey the law. Monthly wages are £300 for policemen, for army officers £500. In Iraq, that's worth risking your life for.

But what of the darkness? What happened to Iraq back in the black days of sectarian war and kidnapping and child-killing? Dr Naji shudders. "I was a fourth year medical student at the forensic medicine institute next to the mortuary and you wouldn't believe what we saw. I remember a body coming in once. It had been decapitated and someone had sewn a dog's head on it. Can you imagine who would do such a thing? It wasn't sewn on very well." Man turned into dog. It's the kind of thing the Safavids would do, or the Mongols. We still need a psychological study of Iraq in 2005-7.

As if sovereignty isn't a problem, the Baghdad government are claiming that the Kurds are selling oil illegally to the Iranians, depriving the government of billions of dollars. Deputy prime minister Hussein Shahristani, a victim of Saddam's torturers – how soon we forget these sufferings of the past, although Iraqis don't – says that the amount is so huge that it is leaving a deficit in the budget.