Optomechanical cooling of multiple degenerate mechanical modes is prevented by the mechanical dark mode due to destructive interference. Here, we report the first experimental demonstration of simultaneous cooling of two near-degenerate mechanical modes by breaking the mechanical dark mode in a two-membrane cavity optomechanical system. The dark mode is generated as the system passes the exceptional point of the anti-parity-time symmetric scheme. By introducing a second cavity mode for the additional dissipative channel, the dark mode is broken and the total phonon number is reduced by more than an order of magnitude below the dark mode cooling limit. Owing to the flexible tunability of the optomechanical coupling rates of such a four-mode coupled system, the optimized cooling efficiency can be achieved by investigating different parameter ranges. Our results provide an important step toward the ground state cooling and entanglement among multiple degenerate mechanical resonators.