BAGHDAD (AP) — A year after it was routed from Iraq in a devastating war that left entire neighborhoods and towns in ruins, the Islamic State group is fighting to hang on to its last enclave in eastern Syria, engaging in deadly battles with U.S.-backed forces.

Cornered in the desert near the Iraqi border with nowhere to run, the militants are putting up a fierce fight, inflicting hundreds of casualties among their opponents and releasing a stream of beheading videos reminiscent of the extremist group's terrifying propaganda at the height of its power.

In Syria, the battle for the area of Hajin...