An electronic device may have a flexible display. The display may have portions that are bent along a bend axis. The display may have display circuitry such as an array of display pixels in an active area and signal lines, thin-film transistor support circuitry and other display circuitry in an inactive area of the display surrounding the active area. The display circuitry may be formed on a substrate such as a flexible polymer substrate. The flexible polymer substrate may be formed by depositing polymer on a support structure that has raised portions. The raised portions may create locally thinned regions in the flexible polymer substrate. The reduced thickness of the flexible polymer substrate in the thinned regions may help ensure that a neutral stress plane that is associated with bending the display along the bend axis is aligned with the display circuitry, thereby minimizing stress in the display circuitry.