Actually, the parts of a turbine engine that bear the strongest thermal load are air cooled indeed. Oil is used to remove heat from the shaft bearings and some accessory systems (engine type dependent), but as mentioned by Blackhand, the combustion chamber and most importantly the turbine section is impossible to cool by oil.
Take a look at the following picture:
The holes in the blade have compressor bleed air flowing out of them that forms a protection layer between the metal and the hot gasses coming from the combustion chamber. If this layer was not there, the exhaust gas would be way too hot for the material to handle. Other, older engines may not use film cooling but simply route cooling air through the blade from foot to tip, but that method is less effective.