QAMISHLI, Syria—This northern city has long been an anomaly where the Syrian regime and a rival Kurdish administration—not quite allies and not quite enemies—coexist while the country’s multisided war rages around them.

But now the government in Damascus is ascendant, having clawed back from the brink of defeat to decimate rebel forces and regain swaths of territory from a weakened Islamic State. Kurds who have made Qamishli a base for the semiautonomous region they have carved out across much of northern Syria during the...