AL-HURRA, Syria—Syrian President Bashar al-Assad, who four months ago seemed on the verge of defeating rebel forces, is now mired in defensive battles on several fronts, complicating efforts to fight the Islamic State militant group.

In a span of weeks, the Islamic State has overrun military bases in Syria's east. In the west, the regime faces a coalition of rebels that threatens the heartland of Mr. Assad's Alawite minority and could alter the course of Syria's multi-sided civil war. Alawites, a Shiite-linked group that forms...