Soon after Turkey officially entered the fray in Syria last month, some 350 Turkish troops marched alongside more than 1,000 U.S.-equipped Syrian rebel forces to clear the Islamic State group out of the Syrian city of Jarablus, north of Aleppo. The battle was over before it began: the IS fighters fled before Turkish tanks rolled in. But the conflict, far from being over, is about to become even more complicated.

With Turkey¡¯s entry into Syria, the conflict there has entered a new phase ¡ª one that may vex the United States, whose partners in the anti-IS coalition already seem more interested in fighting one another. Turkey is not, after all, interested only in clearing IS from its borders; it is also ¡ª and perhaps more ¡ª focused on taking down the Kurds.

To be sure, Turkey does not want IS anywhere near its borders. The group has carried out a series of attacks in Turkey, including a recent suicide bombing in Gaziantep that left 54 dead. But it does not want the Kurdish Democratic Union Party (PYD) there either.