The response of a stably stratified fluid-filled two-dimensional square container to harmonic horizontal oscillations is examined as the forcing amplitude is increased. For the studied forcing frequency, the response flow at very small forcing amplitudes is a synchronous periodic flow with piecewise constant vorticity in regions delineated by the characteristics emanating from the corners of the container, regularized by viscosity (Grayer, et al. 2021). The second temporal harmonic of the forced response flow resonantly excites an intrinsic mode of the stratified container (Thorpe, 1968), whose magnitude grows as the square of the forcing amplitude. Above a critical forcing amplitude, a sequence of pairs of other container modes are excited via triadic resonances with the second-harmonic driven mode. The flows are computed from the Navier--Stokes--Boussinesq equations and the ensuing dynamics are analyzed using Fourier techniques (Buchta, et al. 2021), providing a comprehensive picture of the transition to internal wave turbulence.