Optimal transport (OT) is a widely used tool in machine learning, but computing high-accuracy solutions for large instances remains costly. Entropic regularization and the Sinkhorn algorithm improve scalability; however, when the regularization parameter is small, Sinkhorn convergence slows, and the iterates approach an entropic solution that remains separated from the true OT plan by an entropic-bias plateau. We introduce PINS (Proximal Iterations with sparse Newton and Sinkhorn), a two-loop solver designed to move beyond this plateau. The outer loop applies an entropic proximal-point method, solving the original OT problem through a sequence of entropic subproblems with shifted cost matrices. Each inner subproblem is then solved by a Sinkhorn warm-up followed by sparse-Newton refinement. We prove that PINS converges globally to an optimal solution of the unregularized OT problem and that the inner Hessian admits a sparsification at every outer iteration with a structure independent of the cost matrix. On synthetic and augmented-MNIST instances, PINS achieves much lower relative cost errors than Sinkhorn-type baselines, which stall at the entropic-bias plateau, and is $5$--$73\times$ faster than Sinkhorn with the same outer loop at matched accuracy. On large-scale DOTmark instances, a streaming implementation reduces peak memory by $24$--$54\%$ compared with the network-simplex linear programming (LP) solver and remains feasible under per-process memory budgets for which the LP solver fails.