WASHINGTON—Sen. Elizabeth Warren is proposing a sweeping overhaul to the consumer bankruptcy system that includes removing some much-criticized aspects of a 2005 law supported at the time by the credit-card industry and her Democratic presidential rival Joe Biden.

Ms. Warren is leaning into an issue that catapulted her to the national stage in the 2000s, when she was one of the nation’s leading bankruptcy law experts and a professor at Harvard Law School. The plan, released Tuesday, seeks to make the bankruptcy process cheaper...