Why would Putin want to support Erdogan? Erdogan wants to overthrow Assad and install Saudi/Turk sunni jihadis in Damascus.
1. Putin needs to restore normal relations with Turkey: trade, etc.
2. Turkish military is second largest in NATO and so far has been pro-Western and pro-American. Erdogan is Islamist and wants the military to become ideologically aligned with Islamist forces, not with the west. Putin may not want everything Erdogan wants, but he surely does not want a staunch American ally on his southern border.