Turks (Khazars in Caucasus and Turgesh in north Central Asia) stopped Umayyad Arab hordes after Arabs conquered Iranians.
Turks were they who gave the Arabs their first check to the north and east. In the west only Franks could stop Arabs in central France.

First Turks who became Muslims interacted with Persians, not Arabs.
Muslim Persian Samanid states employed Turkish mercenaries and mamluks, that's how first Muslim Turks appeared. Later these Turks took over the Iranians and created their own empires (Gazhnavids, Karakhanids).

No we aren't from the Arabized south central Asia.

Until 7th century we were in western Mongolia and south Siberia; then we moved to west Kazakhstan where Arabs never ever put foot.

The 'Persian culture' was some Sassanid bureaucratic elements in Seljuk and other Turkic states. Not Arabic.
We didn't have any contact with Arabs until 11th century, which was long after conversion.