In other words, civilized nations are powerless against an uncivilized enemy.
You have to wonder how civilization ever survived.
You don't need to mutilate dead bodies, MOAB women and children or act like Nazis. You merely have to identify your enemy, isolate your enemy and then kill your enemy. That is what war is all about.
Here is how I would handled this Sunni Triangle snake pit:
1. Set up a cordon sanitaire around Fallujah. Nobody goes in. The residents only go out, once, through check points to be sorted into "totally harmeless" or "in need of imprisonment".
2. Wait as long as it takes to have nobody left in the city except the Bathist die-hards.
3. Wage mechanical urban warfare on the city and the holdouts with heavy emphasis on air power and artillery and practically no emphasis on infantry.
4. Sustain the attack until there is not a stray dog left alive to roam the streets.
You have killed your enemy, spared the innocent and taught the enemy in other cities that it is unhealthy to mess with the U.S.
Rinse and repeat as often as necessary.
As long as we rely on local Iraqis for HUMINT, many of these thugs will roam free. It's their home turf. The insurgents are fully aware that HUMINT is our weak link. They've infiltrated the local police and have without doubt acted as phony informers. Not a single Iraqi in that town is to be trusted - yet we need to be able to trust somebody if any kind of precision action is to be taken.