We've been fighting the Arab Sunnis in Iraq. They temporarily allied themselves with Al Qaeda because they thought that alliance could help them obtain their goal, which is the run of the country - just like they had under Hussein. Running out the clock is the strategy they've employed to obtain that goal.
As far as flooding the country with troops at the beginning of the war, we tried that strategy against another insurgency in Vietnam and it didn't work out the way we thought it would. What ended up working is Vietnamizing the war by training ARVN to fight for itself. That's what Creighton Abrams did in Vietnam and that is the strategy that Rumsfeld employed in Iraq with the approval of smart people like General Petraeus. The downside of that strategy is that it takes time to build up a military from scratch. But even if we had been able to rebuild Iraq's military in an instant, it still would have taken time to convince the Sunnis that no matter what they did, they were never going to get the run of the country like they had under Hussein. For the Sunnis to become convinced of that two things had to happen. First, we had to demonstrate our staying power. More importantly though, the Shia had to demonstrate to the Sunnis that there was a very real price to be paid for continuing with the insurgency. The Shia began that demonstration after the Golden Mosque bombing. In effect, after that bombing the Shia began to ethnically cleansed large parts of greater Baghdad. It took about 10 months, but eventually the Sunnis realized that if they continued with the insurgency, they stood a very real chance of being pushed entirely out of Iraq (which is something few Kurds or Shiites would have shed any tears about).
This entire process has taken time to play out and I don't think there is much that could have been done to speed up the process. This is particularly true because what we've been fighting is a Sunni Arab mindset that is not entirely rational. Their belief that they could dominate Iraq with just 15% of the population was ludricous given the fact that the remaining 85% of Iraq was now armed and ready, willing and able to defend themselves. In any event, the Sunnis Arab had to get the stuffing beat out of them not just by us but by the Shia as well before they came to the conclusion the insurgency wasn't going to work.