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.