A display may include an array of organic light-emitting diode display pixels having transistors characterized by threshold voltages subject to transistor variations. Compensation circuitry may be used to measure at transistor threshold voltage for a pixel. The threshold voltage may be sampled by controlling the pixel to sample the threshold voltage onto a capacitor at the pixel. The compensation circuitry may include sense circuitry that may be operated in combination with the pixel to transfer charge from the capacitor to the sense circuitry such that the threshold voltage is produced at an output of the sense circuitry. The compensation circuitry may generate compensation data based on the measured threshold voltage. During display operations, data circuitry may receive digital image data and process the digital image data along with the compensation data to generate analog data signals for the pixel.