The bombing attacks against targets in Fallujah indicate that American forces have obtained good intelligence sources inside Fallujah, otherwise, this campaign of dropping smart bombs against individual buildings (or collections of buildings forming a compound) would not have been tried.
The head of the Iraqi branch of al Qaeda, Abu Musab al-Zarqawi, has sent out an audio tape in which he threatens to assassinate the interim prime minister of Iraq. Both the Baath Party die hards and al Qaeda terrorists want to stop elections and remove the interim Iraqi government. After that, the two mortal enemies would presumably have a death match to decide who would be the new dictator of Iraq.
American troops have developed tactics for working with the new Iraqi government after July 1st. American troops basically provide a back up, and double check, on Iraqi police and security troops. In practice, this means that American troops take care of any of the heavy combat with anti-government forces, but otherwise leave patrolling and policing to the Iraqis. American patrols will come through on a random basis, just to make sure the Iraqi patrols aren't missing anything. These tactics have been used for over a year. At first, the Iraqi force was some tribal or religious militia. But now there are Iraqi police and security troops to work with. A key factor in making this work is staying in touch with the Iraqi forces. This is difficult, because translators are often required, and Iraqi commanders are often not as disciplined and punctual as their American counterparts.