AMIRIYAT FALLUJA, Iraq — One of the Iraqi civilians who risked an escape from the sprawling battle for Falluja made it as far as the Euphrates River. He was there for all to see on Sunday morning: His body, tied to the side of a boat, bobbed in the muddy waters next to a rickety bridge that separates Baghdad from the violence of Anbar Province.

“Sheikh, sheikh, see this man! He drowned,” said a young boy, pointing, as he approached the window of a truck that was slowly crossing the bridge, carrying medicine. “See, see his body.”

The thousands of civilians who managed to flee Falluja and its outskirts and make it to government-controlled areas in recent days faced harrowing journeys, often at night and under fire from Islamic State militants who had been trying to use them as human shields.

Many crossed the wide Euphrates in makeshift boats, and local officials said more than a dozen drowned in the last few days, dying in their own country in the same way that thousands of Syrians and Iraqis have died on the seas trying to reach Europe.