We theoretically investigate the nonlinear optical phenomena including optical bistability and four-wave mixing (FWM) process in a composite photonic-molecule cavity optomechanical system. The photonic-molecule cavity consisted of two whispering gallery mode (WGM) microcavities, where one WGM cavity is an optomechanical cavity with high-cavity dissipation κ and the other WGM cavity is an auxiliary ordinary optical cavity with high-quality factor (Q). Controlling the parameters of the system, such as the coupling strength J between the two cavities, the decay rate ratio δ of the two cavities, and the pump power P, the optical bistability can be controlled. Furthermore, the FWM process which presents the normal mode-splitting is also investigated in the FWM spectrum under different parameter regimes. Our study may provide a further insight of nonlinear phenomena in the composite photonic-molecule optomechanic systems.