Hundreds of foreign women and children linked to Islamic State escaped from a camp in northeastern Syria when Turkish-backed fighters attacked it, U.S.-backed Kurdish forces controlling the area said, raising fears that the Turkish offensive will undermine American efforts to eliminate the extremist group.

The incident occurred at the camp of Ain Eissa, located along a highway Turkey had identified as strategic in the offensive it launched last week to drive Syrian Kurdish forces away from its border.

The...