Multicationic oxide semiconductors are receiving considerable interest in electronic and optoelectronic devices owing to their tunability of physical properties by the cation compositions. Here, we investigated the effects of Ti doping on the electrical properties and gate-bias stability of amorphous zinc–tin–oxide (a-ZnSnO, Zn/Sn = 7:3) thin-film transistors (TFTs) using a cosputtering process. Particularly, by using cosputtering, controllable doping of Ti in a-ZnSnO films was possible in the range of 0.87–3.87 atom %. From various electrical analyses, it was found that the key metrics of Ti-doped ZnSnO (TiZnSnO) TFTs, such as field-effect mobility and gate-bias stability, were highly dependent on the Ti concentration, showing a mobility–stability tradeoff. Based on X-ray photoelectron spectroscopy analysis, the mobility–stability tradeoff is ascribed to the suppression of oxygen vacancy formation by Ti doping. Considering the overall electrical performance and stability of TiZnSnO TFTs, which were processed at 450 °C, the optimal Ti concentration was determined as ∼1.14 atom % with Zn, Sn, and O concentrations of 15.87, 22.92, and 60.07 atom %, respectively. The device exhibited a field-effect mobility of 8.2 cm2 V–1 s–1, a subthreshold slope of 0.208 V decade–1, an on/off ratio of 3.38 × 108, a hysteresis of 2.8 V, and a threshold voltage shift of +5.61 and −2.24 V under positive- and negative-bias stresses, respectively.