Needless provocation from Turkey. Historically, the ancestors of Turkey, the Oghuz Turks, hailed from modern Turkmenistan, east of Aral Sea and far from Xinjiang.
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But modern Turk revivalism efforts attempt to link to an even older Gokturk state which was a loose confederation of nomads on the steppe between Korea and Ukraine.
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And who drove them out? It wasn't any Chinese group. It was other Turkic groups and Mongols.
Oghuz were, mostly, local Turkmen vassals of Gokturks, Khazars, and various other Turkic confederations.