SANLIURFA, Turkey—In Islamic State’s de facto capital of Raqqa, a Syrian city on the banks of the Euphrates, few Syrians hold positions of power these days. Running the show, residents say, are the thousands of foreigners who have converged there to establish an Islamic utopia they believe will soon conquer the planet.

“What we have is a foreign occupation,” said Sarmad al-Jilane, a former electronics student from Raqqa who now runs a website from neighboring Turkey documenting Islamic State abuses in his hometown called...