Depends on which battle they're talking about.
The April 2004, Operation Vigilant Resolve was unsuccessful. Not because we lost on the ground but because we allowed pressure for a non-military (US) solution to be forced on us. Growing out of this was the failed attempt at letting the ING to take control of the city while we forced to stay outside of the city limits. The ING and IP forces were recruited from the local tribes and were heavily infiltrated by insurgents.
Al-Shahwani's current strategy is also fraught with risk. He helped broker the truce that ended the April uprising in Fallujah by setting up the Fallujah Brigade, drawn from the insurgents themselves, to police the restive city. It has left the place a sanctuary for terrorists. Time Magazine
The November 2004 Operation Phantom Fury operation was very successful and ended with the liberation of the city, the only flaw of the operation was that a large portion of the insurgent’s leadership slipped away before the city was locked down.
I was there for Phantom Fury (I got there just after the April battle) and I was back in Fallujah in 2005-2006. The change in the city is amazing.