DETROIT — Detroit’s police chief stepped down Monday after a week of embarrassing revelations about a sexual relationship with a subordinate, forcing the city to search for a fifth leader in four years for a department dealing with one of the nation’s highest violent-crime rates.

Ralph Godbee was hired two years ago to replace a predecessor fired following an affair with a female officer who allegedly also had a relationship with Godbee. His departure extends the revolving door of leadership in a cash-strapped city.