Iran has continually bullied Azerbaijan, which opened the way for Israel to have a relationship with Azerbaijan. Ethnically, the A's are related to the Turks, so the Turks also support Azerbaijan. Turkey and Israel used to have good relations, but Erdogan changed all that. Erdo also found out the Iranians are his primary regional enemy.
Armenia should be an ally of Syria and Iraq as well, but both are pseudostates, and Syria's creaking dictatorship is propped up by the Iranians, with Russians and Turks also in-country. Also, it's not unlikely that most people in the US know someone of Armenian descent (including my favorite neighbors) and would prefer the success of Armenia. Tricky business -- but nations don't have friends and relatives, they have interests.
Turks also hate Kurds. So should the Kurds, the Armenians, Israel and the Azerbaijanis (who have just used Putin to evict Armenians from an energy-rich Armenian enclave so they can build a pipeline) be allies? And can Turkey play, too? Would they want to with the Kurds in the coalition? Would the Armenians want to with the Azerbs and the Turks in the coalition?