An electronic device display may have an array of pixel circuits. Each pixel circuit may include an organic light-emitting diode and a drive transistor. Each drive transistor may be adjusted to control how much current flows through the organic light-emitting diode. Each pixel circuit may include one or more additional transistors such as switching transistors and a storage capacitor. Semiconducting oxide transistors and silicon transistors may be used in forming the transistors of the pixel circuits. The storage capacitors and the transistors may be formed using metal layers, semiconductor structures, and dielectric layers. Some of the layers may be removed along the edge of the display to facilitate bending. The dielectric layers may have a stepped profile that allows data lines in the array to be stepped down towards the surface of the substrate as the data lines extend into an inactive edge region.