An electronic device may be provided having an organic light-emitting diode display and control circuitry for operating the display. The display may include one or more display layers interposed between the control circuitry and a display layer having thin-film transistors. The electronic device may include a coupling structure interposed between the layer of thin-film transistors and the control circuitry that electrically couples the layer of thin-film transistors to the control circuitry. The coupling structure may include a dielectric member having a conductive via, a flexible printed circuit having a bent portion, or a conductive via formed in an encapsulation layer of the display. The display may include a layer of opaque masking material. The layer of opaque masking material may be formed on an encapsulation layer, an organic emissive layer, a thin-film transistor layer, or a glass layer of the organic light-emitting diode display.