Partial differential equations (PDEs) are ubiquitous in science and engineering. Prior quantum algorithms for solving the system of linear algebraic equations obtained from discretizing a PDE have a computational complexity that scales at least linearly with the condition number $\kappa$ of the matrices involved in the computation. For many practical applications, $\kappa$ scales polynomially with the size $N$ of the matrices, rendering a polynomial complexity in $N$ for these algorithms. Here we present a quantum algorithm with a complexity that is polylogarithmic in $N$ but is independent of $\kappa$ for a large class of PDEs. Our algorithm generates a quantum state from which features of the solution can be extracted. Central to our method is using a wavelet basis as an auxiliary basis in which a simple diagonal preconditioner makes the condition number of associated matrices independent of $N$. Crucially, to achieve polylogarithmic complexity, we avoid directly applying this preconditioner. Instead, we introduce a new algorithmic technique that distributes the solution across subspaces of an extended space, yielding a quantum representation from which the desired features can be extracted directly. We present numerical simulations showing the effect of the preconditioner for several differential equations. By removing the condition-number bottleneck, our work restores the possibility of exponential quantum advantage for PDEs and provides a practical way to boost the performance of quantum algorithms that use standard discretization methods.