100,000 troops is just about enough to police the capital of Iraq alone and nowhere near enough to police even 1/2 the country. Even if al-Qaeda in Iraq didn't show up you still would have the massive growth of criminal gangs which would have quickly turned into militias as the Madhi Army did.
Think about it this way, what happened was similar to what would happen if the police force in NY City was disbanded overnight and a foreign police force 1/3rd as big that didn't know the area, the language, or the people was sent in to police NY.
What would happen would be gangs would quickly develop in parts of NY City and turn into militia like super gangs which would basically take over parts of the city and war with each other over parts of the city.
Even without any help from Iran or foreign jihadists major problems with violence were inevitable given the lack of security forces in Iraq which created a massive security vacuum.