This study performs a systematic analysis of the dynamic behavior of the gear for the gear-bearing system with the turbulent flow effect, quadratic damping effect, nonlinear suspension effect, nonlinear oil-film force, and complicated gear mesh force. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents, and fractal dimension of the gear-bearing system. The ignorance of quadratic damping effect for turbomachineries especially in turbulent cases may cause significant errors. The proposed simulation model and theory may provide some useful information for engineers in designing or controlling some turbomachineries particularly in turbulent flow cases.