An electronic device may have a flexible portion that allows the device to be folded. The device may have a flexible display. The flexible display may have a flexible display layer, a cover layer, a touch sensor interposed between the flexible display layer and the cover layer, a support layer, and a polarizer layer. The polarizer layer may be interposed between the touch sensor and the flexible display or between the touch sensor and the cover layer. The touch sensor may include optically clear adhesive that is flexible and facilitates bending of the display. The optically clear adhesive may include additives such as water vapor penetration reducing additive, anticorrosion additive, ultraviolet-light blocking additive, and index-of-refraction adjustment additive. The support layer may be formed from shape memory alloy or amorphous metal and may have openings to facilitate bending.