The unconventional normal-state properties of the cuprates are often discussed in terms of emergent electronic order that onsets below a putative critical doping of x_c ≈ 0.19. Charge density wave (CDW) correlations represent one such order; however, experimental evidence for such order generally spans a limited range of doping that falls short of the critical value x_c, leading to questions regarding its essential relevance. Here, we use X-ray diffraction to demonstrate that CDW correlations in La_2-xSr_xCuO₄ persist up to a doping of at least x = 0.21. The correlations show strong changes through the superconducting transition, but no obvious discontinuity through x_c ≈ 0.19, despite changes in Fermi surface topology and electronic transport at this doping. These results demonstrate the interaction between CDWs and superconductivity even in overdoped cuprates and prompt a reconsideration of the role of CDW correlations in the high-temperature cuprate phase diagram.