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La_2−xSr_xCuO₄ (LSCO) is a prototypical example of hole-doped single layer cuprate superconductor. Here we investigate the low frequency broadband non-linear optical response of LSCO across a wide range of doping as a function of temperature. In addition to the large optical non-linearities that is expected in the superconducting state, we find anomalous non-linear response even in the normal state, well above the superconducting critical temperature. This is in sharp contrast to the clean BCS type superconductors where the strong non-linear response at low temperatures promptly disappear at the critical temperature. Furthermore, we perform 2D THz spectroscopy on these samples to identify the different underlying non-linear processes that give rise to the overall response. We compare our results to existing theories on the superconductivity in this class of materials.