The theory is that without Iraq in 2003, Bush wouldn't have as much of a platform to run on re-election in 2004. His 90% approval rating following 9/11 would have probably dropped to about 40-50% as people begin to realize his ineptness in preventing the attacks before 9/11 and his inability to bring the top leadership of Al Qaeda to justice in a quick manner.
The general theory that he might have less to run on without Iraq has merit, but the assertion that he was inept in preventing the attacks is simply inane. Most of the planning for 9/11 started well back in the Clinton presidency and several of the key participants entered the country, and trained, during Clinton's time. Further, the detection of the terrorist and/or plan was greatly hindered by "the wall" between Law Enforcement and Intelligence that was substantially buttressed under Clinton (you might recall Janet Reno say, "We must build a wall ....").
Testifying before the 9/11 Commission in April of 2004, then-Attorney General John Ashcroft made his own observations about how the “wall” had greatly hindered terrorism investigations:
“In the days before September 11, the wall specifically impeded the investigation into
Zacarias Moussaoui, Khalid al-Midhar and Nawaf al-Hazmi. After the FBI arrested Moussaoui, agents became suspicious of his interest in commercial aircraft and sought approval for a criminal warrant to search his computer. The warrant was rejected because FBI officials feared breaching the wall. When the CIA finally told the FBI that al-Midhar and al-Hazmi were in the country in late August, agents in New York searched for the suspects. But because of the wall, FBI headquarters refused to allow criminal investigators who knew the most about the most recent al Qaeda attack to join the hunt for the suspected terrorists. At that time, a frustrated FBI investigator wrote headquarters, quote, 'Whatever has happened to this – someday someone will die – and wall or not – the public will not understand why we were not more effective and throwing every resource we had at certain problems.'’’
So, no, Bush had not in 8 months completely changed the guidance put in place during 8 years of Clinton, but it takes time to even ID such a problem. Clinton played a large part in creating the problem.
As far as hunting Al Qaeda, several of them were captured fairly quickly, but most worked hard to hide where we couldn't get at them. First, they hid in Taliban Afghanistan, then (once we made it clear we were coming), they ran and hid in extremely remote caves, and then in pseudo-allied Pakistan where they kept an extremely low profile. People with lots of money and lots of fanatical allies can hide pretty well when they want to. The fact that Obama happened to be found during the Obama presidency was irrelevant -- he was found as the culmination of operations put in place years before that just happened to 'get lucky' during his term.