We experimentally study the properties of nonlinear surface gravity waves in a large scale basin. We consider two different configurations: a one-dimensional (1D) monochromatic wave forcing, and a two-dimensional (2D) forcing with bichromatic waves satisfying resonant-wave interaction conditions. For the 1D forcing, we find a discrete wave energy spectrum dominated at high frequencies by bound waves whose amplitudes decrease as a power law of the frequency. Bound waves (e.g. to the carrier) are harmonics superimposed on the carrier wave propagating with the same phase velocity as the one of the carrier. When a narrow frequency random modulation is applied to this carrier, the high-frequency part of the wave energy spectrum becomes continuous with the same frequency-power law. Similar results are found for the 2D forcing when a random modulation is also applied to both carrier waves. Our results thus show that all these nonlinear gravity wave spectra are dominated at high frequencies by the presence of bound waves, even in the configuration where resonant interactions occur. Moreover, in all these configurations, the power-law exponent of the spectrum is found to depend on the forcing amplitude with the same trend as the one found in previous gravity wave turbulence experiments. Such set of bound waves may thus explain this dependence that was previously poorly understood.