An organic light-emitting diode (OLED) display may have an array of organic light-emitting diode pixels that each have OLED layers interposed between a cathode and an anode. Voltage may be applied to the anode of each pixel to control the magnitude of emitted light. The conductivity of the OLED layers may allow leakage current to pass between neighboring anodes in the display. To reduce leakage current and the accompanying cross-talk in a display, a cutting structure may disrupt continuity of the OLED layers. The cutting structure may have two undercuts to disrupt continuity of at least some of the OLED layers. The cutting structure may include a conductive portion that provides a cathode voltage to a cathode layer for the OLED pixels.