Why does Turkey have a variety of tanks? What is the purpose of each type of tank?
Turkey has such a vast array of Cold War (and later eras as well) tanks because they figured out that upgrading existing machines is cheaper than buying new ones. Their M48s, M60s, and Leopard 1s are certainly not the best tanks in the world by any means, however they work just fine in supporting infantry on the low intensity battlefields of the Middle East.
The M48 Pattons of the Turkish army are upgraded with night vision equipment, an improved engine, and a better fire control system. Some are equipped with ERA, I believe.
The Leopard 1A5s of the Turkish Army also have an improved FCS, but no armor upgrades to speak of (unlike Canadian Leo 1s, which had composite armor upgrades).
The M60s of the Turkish Army are unrecognizable from their original form. This is what they looked like when Turkey bought them
This is what they look like now.
The M60 Sabra has a 120mm gun, ERA, composite armor inserts, improved optics and FCS, more powerful engine, and an internal mortar for some reason!
Expensive upgrade, but cheaper than buying new tanks. That being said, Turkey is developing its own tanks, like the Altay, which will be their most modern MBT. When the Altay enters service, the Leopard 2A4s and M60s will replace the M48s and Leo 1s as the second string tanks of the Turkish army (most likely).