Ancient China fought many different steppe empires over thousands of years. You cannot claim all of them as your ancestors simply because they lived in Central Asia.
Shared history does not mean exclusive ownership of history.
Turkey government stops brainwashing your nations. The Great Wall existed centuries before the Göktürk Khaganate was founded in 552 AD.
The Xiongnu, Göktürks, Mongols, and modern Turkish people are not interchangeable historical identities.