Through extensive self-consistent Hartree-Fock calculations in a tight-binding model of twisted bilayer graphene (TBG), we show that many-body effects lead to a considerable increase of the bandwidth of the flat bands and, concomitantly, to a shift of the magic angle (defined by the condition of minimum bandwidth). Specifically, we predict a shift from the textitab initio magic angle of 0.99^∘ to a renormalized value of 0.88^∘ for a TBG sample suspended between metallic gates with a gate-to-gate distance of 10 nm. We derive analytical expressions for the renormalized Fermi velocity and interlayer couplings, finding good agreement with the numerical results, and investigate the convergence toward the numerical solutions with respect to the number of renormalized couplings of a generalized Bistritzer-MacDonald (BM) model. Using the derived analytical formulas, we demonstrate the possibility of tuning the flat bands via different dielectric environments and gate geometries in the experiments. Furthermore, we predict a significant enhancement of the flat-band Fermi velocity at intermediate twist angles relative to the bare value, and propose measurements in this range as a probe of the effective couplings of TBG. Our results imply a change of paradigm whereby the maximum T_c for superconductivity would correspond to a condition of small but not minimum bandwidth.