I recommend reading this book produced by US military, it makes it pretty clear that allies played a minor role in the war overall

a few quotes from it

It would have been much worse if the Iraqi Army had stood and fought. Lots of stories came out from guys at the front who would roll up in Humvees on heavily armoured outposts that were straight up abandoned, where they say there would have been a massacre of us troops if the have stayed. It was such a ubiquitous experience that even 'Generation Kill' has it happen to the protagonists. There was really only one large engagement, and the U.S. Marines absolutely wiped a bunch of Iraqi tanks that were out in the desert, many of whom didn't even get a shot off. All other stuff ended up being insurgency, much of which at the beginning were local militia forces.

What Generation Kill doesn't get into is the theory that the Iraqi commanders and some of the actual troops had been paid off, and also given promises of leniency, which immediately fell through when all former members of the Baathist party were banned from holding government jobs, which put pretty much all the best and brightest of Iraq (including the entire officer corp, most of whom had survived the war) into the position of having to resort to gangsterism (in this form jihadi militia work) to survive in post-war Iraq. While it does make sense that the army wouldn't really want to fight naturally, the rapidity and professionalism of general insurgency only really took off after the Baathists were put completely out of power and points to a broken agreement imo.

i think they mention in Blowback season 1 that pretty much all of the army were either bought off or deserted on their own, taking their guns with them mind you. only in Baghdad there was some serious resistance by the elite Republican Guard.