Twitter Inc. on Wednesday announced the sudden departure of one of its top executives, as it emerges from the shadow of a potential sale and struggles to regain momentum.

The social-media company said Adam Bain would step down as chief operating officer immediately to explore outside opportunities. Mr. Bain’s role will be filled by Twitter’s Chief Financial Officer Anthony Noto, who has been leading the company’s efforts around live video and brokering deals with media companies, including the $10 million deal with the National...