  Electron-boson coupling plays a key role in superconductivity for many
systems. However, in copper-based high-temperature ($T_c$) superconductors, its
relation to superconductivity remains controversial despite strong
spectroscopic fingerprints. Here we use angle-resolved photoemission
spectroscopy to find a striking correlation between the superconducting gap and
the bosonic coupling strength near the Brillouin zone boundary in
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. The bosonic coupling strength rapidly
increases from the overdoped Fermi-liquid regime to the optimally doped strange
metal, concomitant with the quadrupled superconducting gap and the doubled
gap-to-Tc ratio across the pseudogap boundary. This synchronized lattice and
electronic response suggests that the effects of electronic interaction and the
electron-phonon coupling become intimately entangled upon entering the strange
metal regime, which may in turn drive a stronger superconductivity.
