The hybrid squeeze film damper mounted turbulent journal bearing–gear system is proposed in this paper. The nonlinear dynamics of a gear pair supported by such bearing is studied. Numerical results show that, due to the nonlinear factors of lubricant film force, the trajectory of the pinion demonstrates a complex dynamics with dimensionless unbalance parameters. Poincaré maps and bifurcation diagrams are used to analyze the behavior of the pinion trajectory in the horizontal direction. The maximum Lyapunov exponent is used to determine if the system is in a state of chaotic motion. In order to avoid the nonsynchronous chaotic vibrations, an increased proportional gain kp = 0.1 is applied to control this system. It is shown that the pinion trajectory will leave chaotic motion to periodic motion in the steady state under control action.