Magnons, the quanta of spin waves, have significant potential for use in modern technologies, especially when they are strongly coupled to another mode for readout and control. Although magnons strongly interact with microwave photons via the magnetic-dipole interaction to form hybrid cavity-magnon polariton modes, the magnon-phonon interaction in micrometer-sized ferromagnetic spheres is typically weak, which is further aggravated by the large polariton linewidths dominated by the magnon dissipation. The material-limited magnon dissipation rate in particular has been regarded as an unavoidable limitation in these systems. Here, we surpass this long-standing limitation by implementing an active microwave feedback loop to suppress the linewidth of cavity-magnon polaritons and reduce their effective decay rate below the magnon-limited linewidth, thereby enhancing the polariton-phonon cooperativity from C ≃ 1 to C ≃ 300 . As a key milestone, we achieved normal-mode splitting between a cavity-magnon polariton and a mechanical mode, providing direct evidence of three-mode hybridization among photons, magnons, and phonons. Our results establish feedback as a general route to accessing strong-coupling regimes in systems previously thought to be limited by material properties and hence open new opportunities for coherent control in hybrid quantum systems.