Monovalent and divalent aqueous electrolytes confined in negatively charged porous silica are studied by means of molecular simulations including free energy calculations. Owing to the strong cation adsorption at the surface, surface charge overcompensation (overscreening) occurs which leads to an effective positive surface next to the Stern layer, followed by a negatively charged diffuse layer. A simple Poisson-Boltzmann model in which the single-ion potential of mean force is introduced is shown to capture the most prominent features of ion density profiles near an amorphous silica surface. Nevertheless, due to its mean-field nature, which fails to account for correlations, this simple model does not predict overscreening corresponding to charge inversion at the surface. Such an overscreening drastically affects the transport of confined electrolytes as it leads to flow reversal when subjected to an electric field. A simple continuum theory is shown to capture how the electro-osmotic flow is affected by overscreening and by the apparent enhanced viscosity of the confined electrolytes. Comparison with available experimental data is discussed, as well as the implications of these phenomena for ζ-potential measurements.