Charge transport through a randomly oriented multilayered network of two-dimensional (2D) Ti₃C₂T_x (where T_x is the surface termination and corresponds to O, OH and F) was studied using time-of-flight photoconductivity (TOFP) method, which is highly sensitive to the distribution of charge carrier velocities. We prepared samples comprising Ti₃C₂T_x with thickness of 12 nm or 6-monolayers. MXene flakes of size up to 16 μm were randomly deposited on the surface by spin-coating from water solution. Using TOFP, we have measured electron mobility that reached values up to 279 cm²/Vs and increase with electric-field in a Poole-Frenkel manner. These values are approximately 50 times higher than previously reported field-effect mobility. Interestingly, our zero-electric-field extrapolate approaches electron mobility measured using terahertz absorption method, which represents intra-flake transport. Our data suggest that macroscopic charge transport is governed by two distinct mechanisms. The high mobility values are characteristic for the intra-flake charge transport via the manifold of delocalized states. On the other hand, the observed Poole-Frenkel dependence of charge carrier mobility on the electric field is typical for the disordered materials and suggest the existence of an important contribution of inter-flake hopping to the overall charge transport.