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The interplay of superconductivity and charge orders found in all hole-doped cuprates remains an open question in the limit of high H. To address this issue, we have investigated the H-tuned superconducting-normal phase transitions in ultra-high-quality single crystals of La_1.7Eu_0.2Sr_0.1CuO₄ and La_1.48Nd_0.4Sr_0.12CuO₄, which have a striped charge order, under extreme conditions (temperatures T down to 0.016 K and H up to 45 T). A variety of linear and non-linear transport measurements have been performed, and the results reveal a full sequence of states in the T=0 limit as a function of H: a superconductor, a wide regime of 2D superconducting phase fluctuations, and a high-H normal state. Within the phase-fluctuations regime, an unanticipated insulatinglike region emerges that is consistent with a novel type of ordering of localized Cooper pairs within the charge stripes in CuO₂ planes. This regime appears to compete with the H=0 superconductivity. We will present our results and discuss the role of disorder, as well as the similarities to other 2D systems and materials with competing orders that undergo superconducting phase transitions.