A major U.S.-backed assault to break the link between Islamic State’s strongholds in Iraq and Syria began Thursday with Kurdish forces seizing a stretch of highway and moving in on the pivotal Iraqi city of Sinjar, as Americans provided air support and sought to reclaim their influence on the battleground.

The offensive, in which U.S. special operations forces were coordinating airstrikes to back 7,500 so-called Peshmerga fighters, is a strategic gambit to cut a narrow supply line running between Mosul in the east and Raqqa...