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 $n\times n$ linear system that is well-conditioned except for $k$ outlying large singular values in $\tilde{O}(n^{2.065} + k^ω)$ time, improving on a recent result of [Dereziński, Yang, STOC 2024] for all $k \gtrsim n^{0.78}$. 2. We give the first $\tilde{O}(n^2 + {d_λ}^ω$) time algorithm for solving a regularized linear system $(A + λI)x = b$, where $A$ is positive semidefinite with effective dimension $d_λ=\mathrm{tr}(A(A+λI)^{-1})$. 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 $\tilde{O}(n^{2.11})$ time, improving on an $\tilde{O}(n^{2.18})$ 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.