An electronic device display may have pixels formed from crystalline semiconductor light-emitting diode dies, organic light-emitting diodes, or other pixel structures. The pixels may be formed in a display panel having a single substrate or an array of display panel tiles. The display panel has inwardly facing display panel contacts that mate with corresponding outwardly facing interconnect substrate contacts on an interconnect substrate. The interconnect substrate may have areas with compound curvature that are overlapped by the display panel. To enhance flexibility of the interconnect substrate, the interconnect substrate may have flexibility enhancement openings and/or may be formed from a material with a low elastic modulus such as silicone or other elastomeric material.