I don't have the impression that most Americans get just what the European reaction to the Iraq war was like. It was not just that the war was impopular, but that it was widely seen as a crime against humanity and hated by the vast majority of Europeans. To this day the death of Coalition (mainly American) troops in Iraq is prone to elicit responses like 'what were they doing there in the first place?' or even 'seems like they got what they were looking for'.
That's kind of idiotic though. And I say that as a European myself.
Well, I am not endorsing that kind of attitude, but I always have the impression that americans look at Europe's ressentment and think that the war was just impopular and not grasp just how reviled it was.