We consider least-squares problems with quadratic regularization and propose\nnovel sketching-based iterative methods with an adaptive sketch size. The\nsketch size can be as small as the effective dimension of the data matrix to\nguarantee linear convergence. However, a major difficulty in choosing the\nsketch size in terms of the effective dimension lies in the fact that the\nlatter is usually unknown in practice. Current sketching-based solvers for\nregularized least-squares fall short on addressing this issue. Our main\ncontribution is to propose adaptive versions of standard sketching-based\niterative solvers, namely, the iterative Hessian sketch and the preconditioned\nconjugate gradient method, that do not require a priori estimation of the\neffective dimension. We propose an adaptive mechanism to control the sketch\nsize according to the progress made in each step of the iterative solver. If\nenough progress is not made, the sketch size increases to improve the\nconvergence rate. We prove that the adaptive sketch size scales at most in\nterms of the effective dimension, and that our adaptive methods are guaranteed\nto converge linearly. Consequently, our adaptive methods improve the\nstate-of-the-art complexity for solving dense, ill-conditioned least-squares\nproblems. Importantly, we illustrate numerically on several synthetic and real\ndatasets that our method is extremely efficient and is often significantly\nfaster than standard least-squares solvers such as a direct factorization based\nsolver, the conjugate gradient method and its preconditioned variants.\n