We calculate spectral functions of the relativistic <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> model from real-time lattice simulations in classical-statistical field theory. While in the low and high temperature phase of the model, the spectral functions of longitudinal (σ) and transverse (π) modes are well described by relativistic quasi-particle peaks, we find a highly non-trivial behavior of the spectral functions in the cross over region, where additional structures appear. Similarly, we observe a significant broadening of the quasi-particle peaks, when the amount explicit <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> symmetry breaking is reduced. We further demonstrate that in the vicinity of the <math><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math> critical point, the spectral functions develop an infrared power law associated with the critical dynamics, and comment on the extraction of the dynamical critical exponent z from our simulations.