Exponential operator splitting is a very efficient and well established approach for solving time-dependent Schrödinger equations. It is well known that the Lie-Trotter splitting and the Strang-Marchuk splitting converge with order $1$ and $2$, respectively, if the initial data and the potential are sufficiently regular. However, the necessary regularity assumptions for the potential are too restrictive in many applications. In this talk, Strichartz estimates are used to prove new error bounds for exponential splitting methods applied to the linear Schrödinger equation on $\mathbb{R}^d$ (with moderate $d\in\mathbb{N}$). These error bounds show convergence of both methods under weaker assumptions, but the classical orders are reduced by a factor which depends on the regularity of the potential and the dimension $d$. For smooth potentials, the classical orders of convergence are recovered.