A display may have an array of organic light-emitting diodes that form an active area on a flexible substrate. Metal traces may extend between the active area and an inactive area of the flexible substrate. Display driver circuitry such as a display driver integrated circuit may be attached to a flexible printed circuit that is attached to the flexible substrate in the inactive area. The metal traces may extend across a bend region in the flexible substrate. The flexible substrate may be bent in the bend region. The flexible substrate may be locally thinned in the bend region. A neutral stress plane adjustment layer may cover the metal traces in the bend region. The neutral stress plane adjustment layer may include polymer layers such as an encapsulation layer, a pixel definition layer, a planarization layer, and a layer that covers a pixel definition layer and planarization layer.