There are three good reasons why increasing troop strength in Iraq is counterproductive for U.S. interest, to say nothing of anyone else’s.
The first is the current nature of the U.S. military. After the end of the Cold War, structural changes took place to make the U.S. military more flexible for smaller threats in more places as opposed to the set-piece nature of that conflict. That’s why we’ve seen so much use of the reserve and National Guard units. Our military isn’t well set up for long-term occupations to start with; increasing troop strength will only force our brave men and women to play a road game with their weakness against an enemy’s strength. This is not good.
Second, even if there were an increase, we don’t have the stomach to allow them the rules of engagement to get the job done. We just saw the execution of someone whose own rules of engagement were too brutal for Western sensibiities. But his MO was consistent with six millennia of Middle Eastern practice, disgusting as it was. We’re now seen as power holders, and effectively dealing with the challengers Middle East style is just something we won’t do.
Third, even if we did adopt Middle Eastern methods of power holding, we really don’t know which side to favour or crush. Heartless at it sounds, for U.S. interests the current state of conflict in Iraq between Sunni and Sh’ia is probaby the best result it can obtain. The greatest Islamic/Arab weakness is the tendency for them to fight amongst themselves, and given the West’s lack of confidence in its own civilisation and general willingness to fight for it, this is probably as good as it gets.
Americans instinctively always think they have to "do something" about every problem around them. But Christians–who come from a faith where the most important work was done for us–need to recognise that there are problems that, the more we do, the worse it gets. This is one of them, in the land not so far from where our Saviour walked.