I would expect a name from Turkic Altaic mythologies:
Such as Tulpar, Bürküt (Kartal Ata), Bükrek/Buka, Kızagan, Merküt...
Also, there may be a historical reference or name with Turk in it (Gokturk for example), but other than that, is it necessary to put Turk in front of an ordinary bird or...