An electronic device may have a flexible display that overlaps an axis. The display may be supported by a housing. The housing may have first and second portions that rotate relative to each other about the axis. The housing may be placed in an unfolded configuration to support the display in a planar state. The housing may also be placed in a folded configuration by rotating the first and second portions relative to each other. A hinge mechanism may be used to ensure adequate separation between the first and second portions when the housing is bent. Movable flaps may be retracted when the housing is bent to create room for a bent portion of the display.