The conflict between Turkey and the Kurds has escalated since Turkey started a military offensive against the Syrian Kurdish enclave of Afrin on Jan. 20. The Kurdish militia, the People’s Protection Units, which is the United States-led coalition’s top partner in the fight against the Islamic State, controls Afrin. And Turkey is a critical NATO ally.

The Trump administration is floundering. President Recep Tayyip Erdogan of Turkey has ignored President Trump’s appeal to avoid actions that might risk conflict between Turkish and American forces and torpedo the campaign against the Islamic State. Mr. Erdogan has vowed to carry the battle further east to militia-controlled territory stretching all the way to the Iraqi border, where an estimated 2,000 American Special Operations Forces are deployed.

Turkey blames Washington’s support for the People’s Protection Units for the meltdown in American-Turkish ties. Many of the militia’s top cadres are drawn from the Kurdistan Workers’ Party, or P.K.K., the armed group that has been waging a bloody campaign for self-rule inside Turkey. The State Department lists the P.K.K. as a terrorist organization but does not so designate the People’s Protection Units. Turkey insists that the P.K.K. and the militia are the same.

Turkish officials believe that once the Islamic State is defeated, the People’s Protection Units will melt back into the P.K.K. and train its American weapons on Turkey. American officials retort that it was Turkey’s tolerance for — if not outright collusion with — thousands of jihadist fighters who flowed into Syria through Turkey that forced them to embrace the militia. Both arguments have merit.