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. The silicon transistors may be configured in a top gate arrangement. The oxide transistors may be configured in a top gate or a bottom gate arrangement. In one embodiment, source-drain contacts for the silicon and oxide transistors may be formed simultaneously. In another embodiment, the silicon and oxide thin-film transistor structures may be formed using at least three metal routing layers.