Imagine if everything done by the US so far had been done by France. The US would have the same attitude to the problems France would be facing as they have to the US now. This problem has occured many times in history, a country blunders into an occupation and then believes it cannot leave or be forced out or it will lose face. So the policy then becomes finding some way to not look so bad rather than solving the problem. Unfortunately with these situations the only way to look good is to install a puppet government and point to how much they appreciated the invasion.
That usually doesn't work because they don't have the support of the people and that government usually falls when the support is withdrawn. The same thing happened to the Frnch then the US in Vietnam, The British in Iraq, the Russians and now the US in Afghanistan, the British in India, and so on.
If you imagine some other country did all this then you can separate the real issue of solving the problem from the political one of making the leaders who did this look good out of it all. Unless a puppet government (and there won't be any other since the US won't allow anyone but a certain mix of people to runIraq) succeeds soon there is no chance of Bush looking good out of this. You might care about how Bush looks but imagine if Schroeder and Chirac were in Bush's position and it was their prestige at stake.
It's not easy when you vote for someone and they make a mess of things, some people take that personally and a reflection on themselves. Bush is now in dither mode in Iraq, hoping for some excuse or miracle to come along. Historically they almost never do come along, the invader for all his good intentions gets booted out in a humiliating way.
So the unfortunately correct path is to accept the way to go will be humiliating, and fashion the best result possible. Probably the best way is to give the whole mess to the UN because from then on anything that goes wrong can be their humiliation not the US. Later Bush can always say he would have done it differently and blame the UN for not doing it properly. That takes some of the sting out of it.
The other part is to give a lot of money to aid agencies who would be acceptable to the Iraqis, so reconstruction could occur without people resenting it.
The hardest part historically is to remove the face saving part of the equation from solving the problem. People have rarely managed to that. The Russians probably stayed for a long time looking for a face saving solution and had to give up and be humiliated. One face saver is to blame the anti war people as undermining the war effort. The real problem is though in this kind of situation virtually everything becomes part of the humilation, where before dissent was democratic now it becomes insulting to the right wing. Better to have a deep short humiliation than a long drawn out agonising one.