Delayed flight baggage compensation problem

I had delayed baggage, which cost me 300 USD and the airline does not agree to compensate me. The problem is that I bought a 3 leg flight ticket (from EU to Africa) directly from Turkish Airlines, but apparently the last leg was not with them, but Ethiopian Airlines. Turkish Airlines does not want to compensate as the delay was with Ethiopian Airlines and Ethiopian Airlines just simply refuses to compensate. What should I do? I am not sure if I can go to my country's authorities, as Ethiopian Airlines does not fly from my country, but on the other hand I think Turkish airlines should compensate as I bought the whole trip from them