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 sense a current from selected display pixels. A display pixel may include a drive transistor, a gate setting transistor for driving a reference voltage onto the gate terminal of the drive transistor, a data loading and current sensing transistor for connecting the drive transistor to a data/current-sensing line, a light-emitting diode, an emission control transistor coupled between the drive transistor and the diode, and an anode resetting transistor for selectively resetting the anode terminal of the diode. During in-frame current sensing operations, the emission control transistor may be turned off to decouple the drive transistor from the diode, thereby blocking off any residue current and lateral leakage current that may be present at the diode.