GW Bush pursued a very arrogant and unilateral foreign policy, dictating terms to our allies.
Bush followed the ideas of the Project for a New American Century (PNAC). Their basic idea was that now the USSR was out of business, and we found ourselves the last remaining superpower, we ought to act like it. We ought to run the world, like a new Roman Empire. We outlived the other great powers, we should collect the spoils. So he basically told our allies around the world 'You're either for us or against us'.
Our allies had a lot of sympathy for us after 9/11. They wanted to cooperate in a genuine global war against terrorism. Terrorism, they figured, is a global problem, not an American problem. And it requires a global fight. But Bush didn't see it that way.
Bush came to office with an agenda already in place, and his policy was to use every crisis, every emergency, to justify this agenda. Part of it was 'regime change' in Iraq. So Bush tried to bully our allies into a useless and ill-planned invasion of Iraq, even knowing that it had nothing to do with 9/11 or fighting terrorism.
Bush even tried to push the UN around. Since the UN had criticized Saddam, Bush claimed the UN had authorized him to invade Iraq, which was a huge lie. Bush saw the UN as a branch of the US State Department.
So when Obama came to power, our allies breathed a sigh of relief! Obama won a Nobel Peace Prize basically for just replacing Bush!