The quasilinear mixing-length approach to efficient prediction of transport in fusion devices is improved to account for the 'Dimits' upshift between linear and nonlinear critical pressure gradients in zonal-flow-saturated turbulence regimes. This modification uses the frequency mismatch between modes interacting turbulently to track changes in saturation efficiency. Near criticality, energy is transferred exclusively to stable eigenmodes, rapidly increasing the efficacy of the nonlinearity. The modified quasilinear model is able to predict below-threshold turbulent ion-temperature-gradient-driven transport accurately and also yields significantly improved predictions for trapped-electron-mode transport.