Well, not ALL the residents. The "criminal elements" probably regret their loss of power.
I believe the major reason for the turnaround is the training that the IA received from the US military. The Iraqis had no expectations that their own military, or police, for that matter, would ever operate in any mode of 'justice', so they would naturally be suspect of any reconstituted law enforcement. The big difference is how they have been trained; in American methods of treatment of citizens.
The Iraqi citizens saw, in the vast majority of cases, that the Americans would treat them with respect, unless they were suspected of criminal or terrorist activity. They also saw how the criminals and terrorists treated them and their fellow citizens, and realized they'd be a LOT better off under an American model. I think that's when they started turning the bad guys in. They were tired of being punching bags for these terrorists, who were, in most cases, men who were not even Iraqis, but were bringing grief down on their heads by being in their country.