Magnon transport through a magnetic insulator can be controlled by current-biased heavy-metal gates that modulate the magnon conductivity via the magnon density. Here, we report nonlinear modulation effects in 10$\,$nm thick yttrium iron garnet (YIG) films. The modulation efficiency is larger than 40\%/mA. The spin transport signal at high DC current density (2.2$\times 10^{11}\,$A/m$^{2}$) saturates for a 400$\,$nm wide Pt gate, which indicates that even at high current levels a magnetic instability cannot be reached in spite of the high magnetic quality of the films.