Has there ever been a case in modern history of an invading country putting down a local rebellion by force?
England failed in Northern Ireland, The US, French & Chinese all failed in Vietnam, etc, etc...
Actually the only thing that would be close to what the US is trying in Iraq is Vietnam invading Cambodia.
You miss the point.
The United States is in the position to control the Sunni - Shia feud by "balancing the scales". Remember the Battle of Karbala, in 680 AD, was and is sixty miles from Baghdad.
Attempting to "suppress an insurrection" in Iraq is unnecessary. It is not even an insurrection but rather the Shia and Sunni arguing about who is boss. Normal politics, that is. Yeah, bloody by American standards, at least American standards if one ignores 1861 - 65.
By the way, insurrections have been put down by foreign troops time and again. In Arab countries the Turks were notably successful. Philippine insurrection was an American experience. Poland in WWII. Napoleon's troops in Spain. The Normans in the years after 1066. The Hundred Years War. English occupation of French Canada. King Phillip's war.
The French had won in Algeria until the Lefties snatched defeat from victory. Likewise Viet Nam, where the Viet Cong were destroyed during the Tet Offensive.
There are endless examples.