An electronic device may have a flexible display with portions that can be bent. The display may include an array of display pixels in an active area. Contact pads may be formed in an inactive area of the display. Display circuitry in the active area may exhibit a given stack height, whereas display circuitry in the inactive area may exhibit a stack height that is less than the given stack height. In particular, the contact pads may be formed directly on a multi-buffer layer that sits directly on a flexible display substrate. Passivation material may be selectively formed only at the edges of the contact pad on the multi-buffer layer. The multi-buffer layer may be formed at a distance from the edge of the flexible display substrate to minimize cracking in the multi-buffer layer.