An electronic device may have a display. The device and display may bend about a bend axis. The display may have layers such as an organic light-emitting diode layer or other layer with pixels, a touch sensor layer, a protective layer with a polarizer, and a support layer. Lubrication layers formed from textured surfaces, slippery coatings, and lubricants such as oil may be interposed between the layers of the display so that the display layers slip past each other during bending of the device and display and minimize display stress. A device housing may have a recess or other structures that retain the display within the housing while allowing the display layers to shift relative to each other during bending. Elastomeric gaskets and elastomeric lubricant retention structures may be used to mount the display layers in the housing and to retain liquid lubricant within the layers of the display.