Experimental measurements and full-2D numerical simulations show that velocity saturation effects in polysilicon thin-film transistor (TFTs) cannot be neglected in order to obtain a precise modelling of output characteristics. Since full-2D numerical simulations are time consuming and unpractical for circuit simulations, we have developed a new quasi-2D model, that takes into account both velocity saturation effects and the presence of a longitudinal electric field in the Poisson's equation, and includes the effect of parasitic bipolar transistor (PBT) action to reproduce kink effect. The agreement of the quasi-2D model with experimental data from p-channel polysilicon TFTs is very satisfactory even for short channel device, and the presence of a velocity-saturated region with a nearly constant free carrier concentration is reproduced without introducing further assumptions.