A display may be provided with an active central region and a peripheral inactive region. The display may have one or more flexible edges in the peripheral inactive region. Conductive lines may pass between components in the active central region such as display pixels and touch sensor electrodes and components in the inactive peripheral region such as gate driver circuitry and patterned interconnect lines. Each conductive line may have an unbent segment on a portion of a display layer in the active central region and may have a segment on the bent edge of the display layer. The display layer may be formed from a polymer or other flexible material. The bent segments may be configured to be less susceptible to increases in resistance from bending than the unbent segments.