Strong field terahertz pulses are increasingly used to excite and control quantum materials at the ultrafast timescale. They have found widespread application by enabling direct addressing of the superconducting gap or Josephson resonances and are essential in Higgs spectroscopy. Large non-linear optical signals can be induced by the strong coupling of the THz and superconducting degrees of freedom. However, far less attention has been paid to the strong bidirectional coupling between field and material this implies. Here, we use the framework of the time-dependent Ginzburg-Landau equations to study the full field and material evolution of a superconductor driven by strong field terahertz pulses. We find that at high field strengths, the backreaction of the superconductor induces large changes to the driving pulse, which in turn leads to a runaway melting of the superconducting condensate. This results in a surprisingly large sensitivity to the initial driving pulse chirp, enabling these purely dynamical changes to produce order of magnitude differences in the level of melting. We also find large-scale spectral shifting of the driving pulse to occur in just a few hundred nanometers of propagation through a superconductor. We attribute these effects to an inverse plasma redshift, in which the driving field breaks Cooper pairs and decreases the free-electron mobility, analogous to reducing the density of a plasma.