A string of victories by Turkish-backed forces in western Libya this week dealt a heavy blow to the ambitions of the aspiring strongman Khalifa Hifter and signaled the arrival of Turkey as a potentially decisive force among the foreign powers battling for supremacy in the Middle East’s biggest proxy war.

Libyan fighters backed by Turkish firepower captured on Monday a major air base west of Tripoli, the capital, used drones to destroy newly arrived Russian air defense batteries, and on Thursday pressed their offensive by ousting Mr. Hifter’s forces from a key town south of Tripoli.