U.S. oil prices sank deeper into a bear market Tuesday, posting their steepest fall in over three years and a record 12 consecutive days of declines, as fears of oversupply and weakening demand gripped the market.

West Texas Intermediate for December delivery settled 7.1% lower at $55.69 a barrel on the New York Mercantile Exchange, its sharpest one-day fall since September 2015. Brent crude, the global benchmark, was down 6.6% at $65.47, entering a bear market—defined as a 20% drop from a recent peak.

Oil’s...