We have investigated electron transport in side-gated single quantum wires. The wires were produced by deep etching in GaAs/AlGaAs single and quantum well heterostructures. With increasingly negative gate voltage we observed a clear transition from a two-dimensional electron gas (2DEG) to a quasi-one-dimensional electron gas (1DEG). For the first time, enhanced mobility transport is detected in the GaAs/AlGaAs quantum well wire structure whereas a mobility loss occurs in the GaAs/AlGaAs single-heterostructure wire. The controversial behaviour is attributed to electron-density-related screening effects as well as gate-voltage-induced changes of the confinement potential in the growth direction favouring the quantum well wire structure in terms of high-electron-mobility 1DEG transport.