Randomized linear system solvers have become popular as they have the potential to reduce floating point complexity while still achieving desirable convergence rates. One particularly promising class of methods, random sketching solvers, has achieved the best-known computational complexity bounds in theory, but is blunted by two practical considerations: there is no clear way of choosing the size of the sketching matrix apriori; and there is a nontrivial storage cost of the projected system. In this work, we make progress towards addressing these issues by implicitly generating the sketched system and solving it simultaneously through an iterative procedure. As a result, we replace the question of the size of the sketching matrix with determining appropriate stopping criteria; we also avoid the costs of explicitly representing the sketched linear system; and our implicit representation also solves the system at the same time, which controls the per-iteration computational costs. Additionally, our approach allows us to generate a connection between random sketching methods and randomized iterative solvers (e.g., randomized Kaczmarz method). As a consequence, we exploit this connection to (1) produce a stronger, more precise convergence theory for such randomized iterative solvers under arbitrary sampling schemes (i.i.d., adaptive, permutation, dependent, etc.), and (2) improve the rates of convergence of randomized iterative solvers at the expense of a user-determined increases in per-iteration computational and storage costs. We demonstrate these concepts on numerical examples on forty-nine distinct linear systems.