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Quantum phase transitions play an important role in shaping the phase diagram of high-temperature cuprate superconductors. These cuprates possess intertwined orders which interact strongly with superconductivity. However, the evidence for the quantum critical point associated with the charge order in the superconducting phase remains elusive. Here, we reveal the short-range charge orders and the spectral signature of the quantum fluctuations in La_{2-x}Sr_{x}CuO_{4} (LSCO) near the optimal doping using high-resolution resonant inelastic x-ray scattering. On performing calculations through a diagrammatic framework, we discover that the charge correlations significantly soften several branches of phonons. These results elucidate the role of charge order in the LSCO compound, providing evidence for quantum critical scaling and discommensurations associated with charge order.