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Possible mechanisms of overlimiting current in unsupported electrolytes, exceeding diffusion limitation, have been intensely studied for their fundamental significance and applications to desalination, separations, sensing, and energy storage. In bulk membrane systems, the primary physical mechanism is electro-convection, driven by electro-osmotic instability on the membrane surface. It has recently been predicted that confinement by charged surfaces in microchannels or porous media favors two new mechanisms, electro-osmotic flow (EOF) and surface conduction (SC), driven by large electric fields in the depleted region acting on the electric double layers on the sidewalls. Here, we provide the first direct evidence for the transition from SC to EOF above a critical channel height, using in situ particle tracking and current-voltage measurements in a micro/nanofluidic device. The dependence of the over-limiting conductance on channel depth (d) is consistent with theoretical predictions, scaling as d^-1 for SC and d^4/5 for EOF with a transition around d=8um. This complete picture of surface-driven over-limiting current can guide engineering applications of ion concentration polarization phenomena in microfluidics and porous media.