Electron-lattice interactions play a prominent role in quantum materials, making a deeper understanding of direct routes to phonon-mediated high-transition-temperature ($T_{\mathrm{c}}$) superconductivity desirable. However, it has been known for decades that weak electron-phonon coupling gives rise to low values of $T_{\mathrm{c}}$, while strong electron-phonon coupling leads to lattice instability or formation of bipolarons, generally assumed to be detrimental to superconductivity. Thus, the route to high-$T_{\mathrm{c}}$ materials from phonon-mediated mechanisms has heretofore appeared to be limited to raising the phonon frequency as in the hydrogen sulfides. Here we present a simple model for phonon-mediated high-$T_{\mathrm{c}}$ superconductivity based on superfluidity of light bipolarons. In contrast to the widely studied Holstein model where lattice distortions modulate the electron's potential energy, we investigate the situation where lattice distortions modulate the electron hopping. This physics gives rise to small-size, yet light bipolarons, which we study using an exact sign-problem-free quantum Monte Carlo approach, demonstrating a new route to phonon-mediated high-$T_\mathrm{c}$ superconductivity. We find that $T_\mathrm{c}$ in our model generically and significantly exceeds typical upper bounds based on Migdal-Eliashberg theory or superfluidity of Holstein bipolarons. The key ingredient in this bipolaronic mechanism that gives rise to high $T_\mathrm{c}$ is the combination of light mass and small size of bipolarons. Our work establishes principles towards the design of high-$T_{\mathrm{c}}$ superconductors via functional material engineering.

6 pages main text + 12 pages appendices, 5 figures main text + 11 figures appendices