We investigate the effect of disorder on the voltage and layer thickness dependence of the current density in (metal/organic semiconductor/metal) devices containing organic semiconductors with a Gaussian shape of the density of states. The analysis is based on recently published numerically exact expressions for the dependence of the charge-carrier mobility on the carrier density and the electric field in such materials [W. F. Pasveer et al., Phys. Rev. Lett. 94, 206601 (2005)]. For the device simulations, a numerically efficient one-dimensional continuum drift-diffusion device model has been developed, which is also applicable to any other disorder-induced carrier density and field dependence of the mobility and diffusion coefficient. The device and material parameters chosen are relevant to organic light-emitting diode (OLED) applications. It is shown that a realistic degree of disorder can give rise to apparent mobilities that vary over more than 2 orders of magnitude with the layer thickness if the current-voltage curves are (incorrectly) analyzed in terms of the often-used drift-only Mott-Gurney formula. This implies that meaningful analyses of transport in OLEDs should be based on the full functional dependence of the mobility on the carrier density and field, induced by the disorder.