By comparing with recently available experimental data from several groups, we critically discuss the manifestation of continuum many body interaction effects in twisted bilayer graphene (tBLG) with small twist angles and low carrier densities, which arise naturally within the Dirac cone approximation for the non-interacting band structure. We provide two specific examples of such continuum many body theories: one involving electron-phonon interaction and one involving electron-electron interaction. In both cases, the experimental findings are only partially quantitatively consistent with rather clear-cut leading-order theoretical predictions based on well-established continuum many body theories. We provide a critical discussion, based mainly on the currently available tBLG experimental data, on possible future directions for understanding many body renormalization involving electron-phonon and electron-electron interactions in the system. One definitive conclusion based on the comparison between theory and experiment is that the leading order 1-loop perturbative renormalization group theory completely fails to account for the electron-electron interaction effects in the strong-coupling limit of flatband moire tBLG system near the magic twist angle even at low doping where the Dirac cone approximation should apply. By contrast, approximate nonperturbative theoretical results based on Borel-Pade resummation or 1/N expansion seem to work well compared with experiments, indicating rather small interaction corrections to Fermi velocity or carrier effective mass. For electron-phonon interactions, however, the leading-order continuum theory works well except when van Hove singularities in the density of states come into play. (C) 2020 Elsevier Inc. All rights reserved.