  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.
