Those who were able left a long time ago.
The easternmost part of the Sunni-dominated Anbar Province is obviously going to be the front lines of a bloody war between ISIS and the Iraqi government, which is backed by Shi’ite militias. It hasn’t been the easiest place to flee from, with the area around the cities a warzone.
At this point, tens of thousands of civilians in places like Fallujah are just plain stuck in the middle of the offensive, with ISIS urging them to stay put, ISIS fighters putting explosives on the roads into town to try to keep invaders out, and Iraqi forces shelling the same roads to keep ISIS from coming out to meet them.
Even if they could get out, their options are limited. Fleeing deeper into Anbar just means they’re going to be a target of the offensive a little later, and the Iraqi government has viewed Sunni Arab refugees with distrust as possible ISIS infiltrators.
Past offensives into Fallujah, including the US offensives during last decade’s occupation, have been bloody indeed. The Shi’ite militias spearheading the current offensive have a long history of violent recriminations against the civilian population after their offensives as well, leaving the civilians to wait, and worry.