  We study a mechanism to induce superconductivity in atomically thin
semiconductors where excitons mediate an effective attraction between
electrons. Our model includes interaction effects beyond the paradigm of
phonon-mediated superconductivity and connects to the well-established limits
of Bose and Fermi polarons. By accounting for the strong-coupling physics of
trions, we find that the effective electron-exciton interaction develops a
strong frequency and momentum dependence accompanied by the system undergoing
an emerging BCS-BEC crossover from weakly bound $s$-wave Cooper pairs to a
superfluid of bipolarons. Even at strong-coupling the bipolarons remain
relatively light, resulting in critical temperatures of up to 10\% of the Fermi
temperature. This renders heterostructures of two-dimensional materials a
promising candidate to realize superconductivity at high critical temperatures
set by electron doping and trion binding energies.
