A display may have thin-film transistor circuitry that includes organic light-emitting diodes. The thin-film transistor circuitry may be formed on a substrate. First and second thin-film inorganic moisture barrier layers may be deposited on top of the thin-film transistor circuitry. An organic planarization layer may be interposed between the first and second thin-film inorganic moisture barrier layers. A moisture barrier glass layer may be attached to the second thin-film inorganic moisture barrier layer with a layer of liquid adhesive. The display may have functional layers such as a touch sensor and circular polarizer that are interposed between a cover glass layer and the moisture-barrier glass layer. A thermoplastic polymer moisture barrier ring that runs around the peripheral edge of the display may be laser welded between the moisture barrier glass layer and the substrate.