An organic light-emitting diode display may have an array of pixels. Each pixel may have an organic light-emitting diode and thin-film transistor circuitry that controls current flow through the organic light-emitting diode. The thin-film transistor circuitry may include silicon thin-film transistors and semiconducting-oxide thin-film transistors. Double gate transistor structures may be formed in the transistors of the thin-film transistor circuitry. A double gate transistor may have a semiconductor layer sandwiched between first and second dielectric layers. The first dielectric layer may be interposed between an upper gate and the semiconductor layer and the second dielectric layer may be interposed between a lower gate and the semiconductor layer. Capacitor structures may be formed from the layers of metal used in forming the upper and lower gates and other conductive structures.