An electronic device may be provided with a display having a flexible substrate with bent edges. The flexible substrate may have a planar active region that includes an array of light-emitting elements such as organic light-emitting diodes with associated control lines. The flexible substrate may also have inactive regions that lie outside of the active region. The bent edges may be formed from portions of the flexible substrate in the inactive regions. Traces for distributing control signals to the control lines in the active region may be formed in the inactive regions. Corner openings may be formed at the corners of the flexible substrate to accommodate bending of the flexible substrate in the inactive regions. A jumper or a portion of the flexible substrate that lies outside of a corner opening may be used to convey signals between traces on adjoining inactive regions.