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 — 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 — and perhaps more — 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.