The propagation of fast electron currents in near solid-density media was investigated via proton probing. Fast currents were generated inside dielectric foams via irradiation with a short (∼0.6 ps) laser pulse focused at relativistic intensities (Iλ^{2}∼4×10^{19} W cm^{-2} μm^{2}). Proton probing provided a spatially and temporally resolved characterization of the evolution of the electromagnetic fields and of the associated net currents directly inside the target. The progressive growth of beam filamentation was temporally resolved and information on the divergence of the fast electron beam was obtained. Hybrid simulations of electron propagation in dense media indicate that resistive effects provide a major contribution to field generation and explain well the topology, magnitude, and temporal growth of the fields observed in the experiment. Estimations of the growth rates for different types of instabilities pinpoints the resistive instability as the most likely dominant mechanism of beam filamentation.