An electronic device may have a display overlapped by an image transport layer such as a coherent fiber bundle or layer of Anderson localization material. The image transport layer may have an input surface that receives an image from the display and a corresponding output surface to which the image is transported. The input surface and output surface may have different shapes. During fabrication of the image transport layer, a peripheral portion of the image transport layer may be laterally indented to form a peripheral recess. Electrical components such as optical and radio-frequency components may be mounted in the recess and may be overlapped by peripheral portions of the image transport layer and/or may be mounted under the display.