Turkey’s unwillingness to intervene in the battle over a predominantly Kurdish Syrian city on its border has earned the country harsh criticism from Washington, exposing a dangerous rift over how the two allies want to tackle Islamic State’s rise.

After more than three weeks of fighting for Kobani and its surroundings, the extremist group edged closer to seizing the city from Syrian Kurdish militia fighters on Thursday. By nighttime, city officials said the militants had managed to gain control of about a quarter of Kobani...