They have organised elections, and pushed through a new democratically-ratified constitution that has given birth to a national government with a true mandate. They have sent more of their own troops, and trained the locals. They have sacrificed some 2,700 of their servicemen and over $300bn (£1.6bn) of their taxpayers' money. But nothing the Americans can do has stopped post-Saddam Iraq's long slide into chaos and despair.
President Bush and his top aides still insist civil war has not broken out. That, however, is a matter of semantics after the latest UN report that almost 6,600 people died in sectarian violence in the last two months for which statistics are available - an "unprecedented" 3,590 in July, followed by 3009 in August.
That is 108 killings every day. As Kofi Annan, the UN Secretary General, put it this week during the final General Assembly of his tenure: "If current patterns of alienation and violence persist much longer, there is a grave danger that the Iraqi state will break down, possibly in the midst of full-scale civil war."
That was not what the Bush administration intended when Nuri al-Maliki became prime minister four months ago. The security situation has only worsened since and basic services have become ever more intermittent. Even the most upbeat US commanders admit, if Baghdad cannot be saved, Iraq is lost.
That fate, according to a leaked recent report by the Marines' top intelligence officer in Iraq, has befallen Anbar province, home of the "Sunni Triangle" - which accounts for nearly a third of the country.
The report, which has not been seriously disputed by the Pentagon, concludes that nothing US forces at their present levels can do in Anbar will bring it under control. The writ of the central government does not run there. In Baghdad too, nothing seems to work. Earlier this summer Mr Malaki announced a massive security blitz in the capital by US and Iraqi troops.
The upshot has been only greater violence in a city where private militias rule entire neighbourhoods. Now there is talk of a fortified cordon around the city - but scant prospect that this initiative will succeed.
Not surprisingly, muttering here against Mr Maliki grows more audible by the day. President Bush was being "driven crazy" by Mr Maliki's indecisiveness, The New York Times reported on Wednesday, quoting a former senior US official. The pictures of a smiling Mr Maliki on his recent visit to Iran, apparently getting on famously with Washington's current nemesis, Mahmoud Ahmadinejad, cannot have improved President Bush's humour.
The killings spare no one. Pentagon figures show that attacks between May and August this year were running at 792 a week, up 24 per cent from the previous quarter. Sectarian violence is increasing, but so too are attacks on US troops.
The hope was to reduce US forces in Iraq in this US election year. Instead they have been increased, from 127,000 in January to 144,000 now. Each day, two or three more American soldiers die. Since the March 2003 invasion over 2,690 have been lost.
Three decades ago, a decorated young veteran who would become the Democratic presidential candidate in 2004 famously asked: "How do you ask a man to become the last man to die for a mistake?" The words of John Kerry to a Congressional Committee that was probing the Vietnam war now apply to Iraq - to American soldiers and Iraqi civilians alike.Reuse content