This study focuses on the interaction between the water free surface, the riverbed, and some Darrieus-type hydrokinetic turbines deployed in river flows. As turbines offer a resistance to the flow, they affect the upcoming velocity, which in turn affects their performance. The proximity of the neighboring deformable free surface or rigid bed may also influence their power extraction. In this context, 2D and 3D URANS simulations of a cross-flow (H-Darrieus type) turbine are conducted with free-surface modeling and adapted boundary conditions allowing the capture of the interactions between the turbine and the resource. Different water depth immersions are considered in order to study local proximity effects. It is found, neglecting riverbed friction, that shallow immersion is detrimental to power extraction whereas bed proximity associated with deep immersion is favorable. This observation does not hold when considering a more realistic river with a velocity profile throughout the depth. Direction of rotation in high proximity cases also plays a role. Although the literature suggests a slight increase in power extraction with the Froude number, we find that when interaction with the resource is taken into account, the power extraction is rather independent of the Froude number for deep immersion or slightly decreasing for shallow immersion. Nonetheless, all the variations in power extraction reported in this study remain small compared to the ones associated with blockage effects. Finally, the shallow immersion case simulated in 3D behaves similarly to that simulated in 2D. Switching the orientation of the rotation axis from horizontal to vertical, despite changing the local interaction with the free surface, does not affect significantly the performance of the turbine.