Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finite-dimensional systems, less is known about quantum algorithms for real-space dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a $d$-dimensional Schrödinger equation with $η$ particles can be simulated with gate complexity $\tilde{O}\bigl(ηd F \text{poly}(\log(g'/ε))\bigr)$, where $ε$ is the discretization error, $g'$ controls the higher-order derivatives of the wave function, and $F$ measures the time-integrated strength of the potential. Compared to the best previous results, this exponentially improves the dependence on $ε$ and $g'$ from $\text{poly}(g'/ε)$ to $\text{poly}(\log(g'/ε))$ and polynomially improves the dependence on $T$ and $d$, while maintaining best known performance with respect to $η$. For the case of Coulomb interactions, we give an algorithm using $η^{3}(d+η)T\text{poly}(\log(ηdTg'/(Δε)))/Δ$ one- and two-qubit gates, and another using $η^{3}(4d)^{d/2}T\text{poly}(\log(ηdTg'/(Δε)))/Δ$ one- and two-qubit gates and QRAM operations, where $T$ is the evolution time and the parameter $Δ$ regulates the unbounded Coulomb interaction. We give applications to several computational problems, including faster real-space simulation of quantum chemistry, rigorous analysis of discretization error for simulation of a uniform electron gas, and a quadratic improvement to a quantum algorithm for escaping saddle points in nonconvex optimization.