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 pixel conditioning and current sensing operations, the anode resetting transistor may be constantly turned on to ensure that there is no voltage perturbation at the source terminal of the drive transistor, which can help prevent hysteresis-induced current sensing error.