We compute the far-from-equilibrium dynamics of relativistic scalar quantum fields in $3+1$ space-time dimensions starting from over-occupied initial conditions. We determine universal scaling exponents and functions for two-point correlators and the four-vertex in a self-similar regime in time and space or momenta. The scaling form of the momentum-dependent four-vertex exhibits a dramatic fall-off toward low momenta. Comparing spectral functions (commutators) and statistical correlations (anticommutators) of field operators allows us to detect strong violations of the fluctuation-dissipation relation in this non-perturbative infrared regime. Based on a self-consistent expansion in the number of field components to next-to-leading order, a wide range of interaction strengths is analyzed and compared to weak-coupling estimates in effective kinetic theory and classical-statistical field theory.