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High-T_c cuprates exhibit a rich phase diagram featuring intertwined density wave orders and superconductivity. It is crucial to unravel the interactions between different orders to understand these important materials. Combining x-ray free electron laser with a pulsed magnet, we elucidate these interactions in a prototypical cuprate La_1.885Sr_0.115CuO₄, by studying the charge density wave (CDW) and effects of large magnetic fields (H) up to 24 Tesla. We show the CDW peaks reveal two distinct regions in the material: a majority phase with short-range CDW coexisting with superconductivity, and a minority phase with longer-range CDW coexisting with static spin density wave (SDW). At low magnetic fields, the CDW intensity grows smoothly in a manner like the SDW. However, at high fields upon entering the vortex-liquid state, we discover a sudden increase in the CDW amplitude. Our result signifies a strong response of the CDW to mobile superconducting vortices, and further suggests an intimate connection between enhanced CDW amplitude and local superconducting pairing across the H-T phase diagram.