Per Wikipedia, Kurds are a diverse group, embracing a variety of beliefs ranging from mild or mystical forms of Islam to pagan practices to Christianity. From the lack of reports, I would infer there isn’t much internal strife over matters of faith among Kurds.
The Kurds are one of the largest ethnic groups without a nation of their own.
Turkey may be encouraging “Kurdistan” to push Kurds in Turkey to, easing the ethnic strife they otherwise cause.
Or it may be supporting Kurdistan so that the Kurds fight ISIS in Iraq, instead of the war spreading to Turkey.
Or Turkey may be supporting the Kurds to get the oil in an increasingly turbulent time - and not having to rely on Russia like its European neighbors.