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We present a new class of preconditioned iterative methods for solving linear systems of the form Ax=b. Our methods are based on constructing a low-rank Nyström approximation to A using sparse random matrix sketching. This approximation is used to construct a preconditioner, which itself is inverted quickly using additional levels of random sketching and preconditioning. We prove that the convergence of our methods depends on a natural average condition number of A, which improves as the rank of the Nyström approximation increases. Concretely, this allows us to obtain faster runtimes for a number of fundamental linear algebraic problems: 1. We show how to solve any n×n linear system that is well-conditioned except for k outlying large singular values in O~(n^2.065+k^ω) time, improving on a recent result of [Dereziński, Yang, STOC 2024] for all k≳n^0.78. 2. We give the first O~(n^2+d_λ^ω) time algorithm for solving a regularized linear system (A+λI)x=b, where A is positive semidefinite with effective dimension d_λ=tr(A(A+λI)^{−1}). This problem arises in applications like Gaussian process regression. 3. We give faster algorithms for approximating Schatten p-norms and other matrix norms. For example, for the Schatten 1-norm (nuclear norm), we give an algorithm that runs in O~(n ^{2.11}) time, improving on an O~(n ^{2.18}) method of [Musco et al., ITCS 2018]. All results are proven in the real RAM model of computation. Interestingly, previous state-of-the-art algorithms for most of the problems above relied on stochastic iterative methods, like stochastic coordinate and gradient descent. Our work takes a completely different approach, instead leveraging tools from matrix sketching.