So you contend that our anti-terrorism strategy should consist of plopping down hundreds of thousands of troops in the middle of the Islamic "bee hive" and keeping them there indefinitely, so as to distract the terrorists and keep them busy killing our soldiers instead of our civilians. How many decades and how many lives do you think it might take before this brilliant strategy exhausts the global supply of Islamic radicals? How long might it be before the terrorists get smart, grow weary of killing our soldiers in their backyard, and return to killing civilians in America? How long might it be before this strategy ends up with us having bitten off more than we can chew?
In a word, yes. If a young jihadi in, say, Saudi Arabia or Iran wants to go fight the Great Satan today, where does he go? Well, clearly, he goes to Iraq, because the media is telling him that the US is losing to his buddies. He thinks he can go to Iraq and kill the soldiers of the Great Satan. In reality, our soldiers are mostly killing the terrorists, not the other way around. The basic answer to your question is, yes, this is a smart strategy. We attract terrorists from around the world to Iraq precisely because they *think* they can hurt us there. The truth is, they mostly end up getting killed. If we don't provide that battleground then the terrorists come over here, and *we* mostly end up getting killed. On September 11, a mere 19 terrorists killed around 3,000 Americans. In Iraq, any given 3,000 terrorists are probably lucky to kill 19 Americans. I like those numbers quite a bit better, although any American deaths are regrettable. So yes, I'd much prefer that the young jihadi go up against our soldiers rather than spend his time looking for our women and children at home and around the world.
Is it perfect? Definitely not. Is it better than your strategy of hiding in a hole and hoping the terrorist world forgets about us? Yep.