An electronic device may be provided with a display such as an organic light-emitting diode display. The display may include an array of display pixels formed on a polymer substrate layer. The polymer substrate layer may include an contiguous layer of polyimide that forms a substrate layer in additional structures such as a polymer film and a flexible printed circuit. A first transition region may be interposed between the display and the polymer film, and a second transition region may be interposed between the polymer film and the flexible printed circuit. Metal traces may be formed on the polymer film and on the flexible printed circuit. A display driver integrated circuit may be mounted to the traces on the polymer film. The polymer film may form a U-shaped bend. The flexible printed circuit may be coupled to a printed circuit board in the device using hot bar solder connections.