If they do well and exterminate the fascists Saddamites amongst them, and if we give unwaivering support .... then it will spread to other cities and locals in the north and south as well.
I assume you are referring to the destruction of the Baath party, and I agree with that. However, don't forget that the shiites are the majority in Iraq. They dominate the population throughout southern Iraq. For decades, they have been politically dominated by the sunni muslims, who are a minority overall, but who form the majority in central Iraq, including Baghdad. Saddam is a sunni.
It would not be good to have the shiites start 'exterminating' sunnis in general. For one thing, it might make a popular uprising in Baghdad less likely. Secondly, the shiites are more fundamentalist than the sunnis and are a natural ally to Iran.
One of the tightropes we will have to walk after this is over, is to fashion a post-saddam Iraq which remains independent from and somewhat of a balance to Iran.