We develop a nonlinear theory of the electronic transport in superconductors in the framework of the time-dependent Ginzburg-Landau (TDGL) equation. We utilize self-consistent Gaussian approximation and reveal the conditions under which the current-voltage V(I) dependence (I-V characteristics) acquires an S-shape form leading to switching instabilities. We demonstrate that in two-dimensions the emergence of such an instability is a hallmark of the Berezinskii-Kosterlitz-Thouless (BKT) transition that we have detected by transport measurements of titanium nitride (TiN) films. Our theoretical findings compare favorably with our experimental results.