We study the normal-state, doping-driven phase diagram of the square-lattice Hubbard model using the dynamical cluster approximation combined with the numerical renormalization group as a cluster solver, which gives direct access to real-frequency dynamics at essentially zero temperature. In a parameter regime relevant for cuprates, U=7t and t'=-0.3t, we find a critical doping p^∗ that marks a continuous quantum phase transition between a pseudogap metal and a normal Fermi liquid. The transition is identified by a continuous collapse, from both sides, of the Fermi-liquid scale extracted from charge, spin, and d_(x²-y²)-wave pairing susceptibilities. This collapse produces a non-Fermi-liquid regime at intermediate energy scales, which appears to extend to arbitrarily low scales at p^∗. As p^∗ is crossed from the normal Fermi liquid at p>p^∗ into the pseudogap metal at p<p><p^∗, since the zero-frequency spectral weight remains large in the nodal region but is strongly suppressed in the antinodal region.