There were Shiite Turkic confederations -- the Ottomans and other beyliks, however, were thoroughly Sunni.
If, in the post-Timurid aftermath, the Ak or Qara Koyunlu conquer the Anatolian beyliks a la the Safavids, you might be able to have Shiite Anatolia.
Would An Turkic Shiite Empire be able to become that large and expand into Arabia, ruling Sunni Arabs ? How would Persia develope, since they didn´t become really Shiite until 18th century ?