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, the pixel definition layer may disrupt continuity of the OLED layers. The pixel definition layer may have a steep sidewall, a sidewall with an undercut, or a sidewall surface with a plurality of curves to disrupt continuity of the OLED layers. A control gate that is coupled to a bias voltage and covered by gate dielectric may be used to form an organic thin-film transistor that shuts the leakage current channel between adjacent anodes on the display.