Phonon anharmonicity plays a crucial role in determining the stability and vibrational properties of high-pressure hydrides. Furthermore, strong anharmonicity can render phonon quasiparticle picture obsolete questioning standard approaches for modeling superconductivity in these material systems. In this work, we show the effects of non-Lorentzian phonon lineshapes on the superconductivity of high-pressure solid hydrogen. We calculate the superconducting critical temperature TC ab initio considering the full phonon spectral function and show that it overall enhances the TC estimate. The anharmonicity-induced phonon softening exhibited in spectral functions increases the estimate of the critical temperature, while the broadening of phonon lines due to phonon-phonon interaction decreases it. Our calculations also reveal that superconductivity emerges in hydrogen in the C m c a - 12 molecular phase VI at pressures between 450 and 500 GPa and explain the disagreement between the previous theoretical results and experiments.|This work studies the effects of non-gaussian phonon lineshapes from stochastic self-consistent harmonic approximation on the superconducting critical temperature (Tc) of hydrogen at high pressure. It predicts superconductivity in the Cmca-12 phase between 450 and 500 GPa and an increase in Tc for both the Cmca-12 and the I41/amd-2 structures compared to harmonic calculations.