With the arrival in Iraq of the fifth and final reinforcement brigade, American forces have now taken the fight against extremist militants beyond Baghdad, striking at al-Qaeda positions in the volatile Diyala province. At least 30 militants have been killed in the past two days and huge quantities of weapons seized. The fighting comes amid a new upsurge of violence: some 87 people were killed when al-Qaeda attacked a Shia mosque in Baghdad on Tuesday, an atrocity that sparked three retaliatory attacks on Sunni mosques south of the city. Across the country, suicide bombings, shootings and sectarian murders have continued, with 33 people found shot and abandoned in Baghdad and coalition forces killing or arresting dozens of suspected insurgents.

The Americans have given warning of a “summer of hard fighting”. And in his interview with The Times, General David Petraeus predicted that the surge in US forces – which has added 28,000 to the total deployment – will provoke al-Qaeda into a new bombing campaign to prove that it is far from beaten. Al-Qaeda, however, has already suffered a serious setback in several key provinces, especially in Anbar, where Sunni tribesmen, sickened by the extremism of these foreign militants, have turned against them. Al-Qaeda can no longer pose as the spearhead of an insurgency against foreign occupation or the defenders of the Sunni minority: its tactics have brought only bloodshed and retaliation, and many indigenous Sunnis are now ready to help coalition forces in hunting them down.

General Petraeus gives explicit credit to this operation to turn Iraqi tribes against al-Qaeda to British commanders. The British forces had been “hugely useful” in strengthening the coalition with diplomatic skills and experience learnt in Northern Ireland. And he made a strong appeal to the British Government not to pull out early from Iraq but to stay the course in the south.

His assessment should carry weight – with Gordon Brown as well as in Washington. The general is now in his fourth tour of duty and is by far the most experienced and competent US commander in the four years since the Iraq war. He is due to report to Congress in September, and is determined to demonstrate significant military and political progress. The second point, he told The Times, is by far the most important: operations on the ground can only buy time for the Government of Nouri al-Maliki, which must understand the need to reach out to the Sunnis, work for political reconciliation and, especially, to agree a deal on sharing oil revenues.

Much depends on the current operations against al-Qaeda. The four-month security crack-down is focused on Baghdad and the pressure is beginning to pay off. The Americans are speaking of co-ordinated operations throughout all the provinces – and here again British support will be vital. So, too, is political will in Washington. Iraq is one of the most divisive political issues in America, and in his interview with The Times Fred Thompson, the US actor fast emerging as a leading conservative Republican candidate, made clear that he favours giving the surge more time to work. It will take political courage to stand against popular calls for an early pull-out. If Washington and London stand behind General Petraeus, he may yet overcome al-Qaeda as well as lift the West’s despair over Iraq.