JERUSALEM—The Trump administration won’t withdraw forces from northern Syria unless Turkey offers a firm commitment not to target the U.S.’s Kurdish allies, White House national security adviser John Bolton said Sunday.

President Trump’s order to withdraw U.S. troops is a “cause-and-effect mission” that requires certain assurances from various players in the region before it can be executed, said Mr. Bolton, the first administration official to outline the conditions for withdrawal.

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