We show that an insulated electrostatic gate can be used to strongly suppress ubiquitous background charge noise in Schottky-gated GaAs/AlGaAs devices. Via a 2D self-consistent simulation of the conduction band profile we show that this observation can be explained by reduced leakage of electrons from the Schottky gates into the semiconductor through the Schottky barrier, consistent with the effect of "bias cooling." Upon noise reduction, the noise power spectrum generally changes from Lorentzian to 1/f type. By comparing wafers with different Al content, we exclude that DX centers play a dominant role in the charge noise.