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Relaxation of linear magnetization dynamics is well described by the viscous Gilbert damping processes. However, for strong excitations, nonlinear damping processes such as the decay via magnon-magnon interactions emerge and trigger additional relaxation channels. Here, we use space- and time-resolved microfocused Brillouin light scattering spectroscopy and micromagnetic simulations to investigate the nonlinear relaxation of strongly driven propagating spin waves in yttrium iron garnet nanoconduits. We show that the nonlinear magnon relaxation in this highly quantized system possesses intermodal features, i.e., magnons scatter to higher-order quantized modes through a cascade of scattering events. We further show how to control such intermodal dissipation processes by quantization of the magnon band in single-mode devices, where this phenomenon approaches its fundamental limit. Our study extends the knowledge about nonlinear propagating spin waves in nanostructures which is essential for the construction of advanced spin-wave elements as well as the realization of Bose-Einstein condensates in scaled systems.