We study the relaxation dynamics of a trapped polariton gas in the nonlinear regime. We excite the three lowest energy states of the system and observe the time evolution of the polariton density in the momentum space. At a low excitation power, the dynamics is characterized by dipole oscillations of constant amplitude. A damping of these oscillations is observed at a high excitation power. It is attributed to collisional relaxation within the coherent polariton gas. We investigate the dependence of this effect on the excitation power, polarization, and polariton excitonic content to highlight the role of polariton-polariton scattering. The experiments are described in the frame of a Gross-Pitaevskii mean-field theory. We find a good agreement between the theoretical simulations and the experimental observations. Analysis of the theoretical model reveals that multiple parametric scattering and final-state stimulation are responsible for the damping of the oscillations.