A display may have an array of organic light-emitting diode display pixels. Each display pixel may have a light-emitting diode that emits light under control of a thin-film drive transistor. Each display pixel may have thin-film transistors and capacitor structures that form a circuit for compensating the drive transistor for threshold voltage variations. The capacitor structures may be formed from interleaved stacked conductive plates. The conductive plates may be formed from layers of material that are used in forming the drive transistor and other thin-film transistors such as a semiconductor layer, a first metal layer, a second metal layer, a third metal layer, and interposed dielectric layers.