French President Emmanuel Macron is facing a stormy political year, having appointed his third prime minister in just 12 months. On Friday, Macron named Francois Bayrou, a seasoned centrist politician, as the new head of government after a tense two-hour meeting.

Bayrou, 73, is stepping into a highly unstable political climate. A former presidential candidate and long-time ally of Macron, Bayrou is tasked with leading France out of its growing political and financial troubles. But can he...