An electronic device may include a display having an array of display pixels on a substrate. The display pixels may be organic light-emitting diode display pixels or display pixels in a liquid crystal display. In an organic light-emitting diode display, hybrid thin-film transistor structures may be formed that include semiconducting oxide thin-film transistors, silicon thin-film transistors, and capacitor structures. The capacitor structures may overlap the semiconducting oxide thin-film transistors. Organic light-emitting diode display pixels may have combinations of oxide and silicon transistors. In a liquid crystal display, display driver circuitry may include silicon thin-film transistor circuitry and display pixels may be based on oxide thin-film transistors. A single layer or two different layers of gate metal may be used in forming silicon transistor gates and oxide transistor gates. A silicon transistor may have a gate that overlaps a floating gate structure.