An electronic device may have pixels that form an active area for a flexible display layer. An inactive area that is free of pixels may surround the active area. One or more edges of the flexible display layer may be bent around respective bend axes. The display layer may also have one or more unbent edges in which the inactive area and active area of the display have a planar shape. An image transport layer may have an input surface that receives the image and an output surface to which the image is transported. The image transport layer may be formed from a coherent fiber bundle or a layer of Anderson localization material. The coherent fiber bundle may have straight fibers that run parallel to each other in a vertical direction that is parallel to a surface normal of a central portion of the display layer.