Application of a gate bias to an organic field-effect transistor leads to accumulation of charges in the organic semiconductor within a thin region near the gate dielectric. An important question is whether the charge transport in this region can be considered two-dimensional, or whether the possibility of charge motion in the third dimension, perpendicular to the accumulation layer, plays a crucial role. In order to answer this question we have performed Monte Carlo simulations of charge transport in organic field-effect transistor structures with varying thickness of the organic layer, taking into account all effects of energetic disorder and Coulomb interactions. We show that with increasing thickness of the semiconductor layer the source-drain current monotonically increases for weak disorder, whereas for strong disorder the current first increases and then decreases. Similarly, for a fixed layer thickness the mobility may either increase or decrease with increasing gate bias. We explain these results by the enhanced effect of state filling on the current for strong disorder, which competes with the effects of Coulomb interactions and charge motion in the third dimension. Our conclusion is that apart from the situation of a single monolayer, charge transport in an organic semiconductor layer should be considered three-dimensional, even at high gate bias.