A display may have organic light-emitting diode pixels formed from thin-film circuitry. An organic layer including planarization layers and a pixel definition layer may overlap the thin-film circuitry. Thin-film encapsulation may overlap the organic layer. The thin-film encapsulation may be formed from an organic dielectric layer interposed between two layers of inorganic dielectric material. A strip of peripheral crack-stopper structures may run along an edge of the display and may surround the array of pixels. The crack-stopper structures may include parallel inorganic lines formed from a first inorganic layer such as an inorganic layer of the thin-film circuitry. A strip of the organic layer may overlap the parallel inorganic lines. The crack-stopper structures may have parallel tapered polymer lines. The polymer lines may be overlapped by a second inorganic dielectric layer formed from the inorganic material of the thin-film encapsulation layer.