We show that the shadow vertex simplex algorithm can be used to solve linear programs in strongly polynomial time with respect to the number $n$ of variables, the number $m$ of constraints, and $1/δ$, where $δ$ is a parameter that measures the flatness of the vertices of the polyhedron. This extends our recent result that the shadow vertex algorithm finds paths of polynomial length (w.r.t. $n$, $m$, and $1/δ$) between two given vertices of a polyhedron. Our result also complements a recent result due to Eisenbrand and Vempala who have shown that a certain version of the random edge pivot rule solves linear programs with a running time that is strongly polynomial in the number of variables $n$ and $1/δ$, but independent of the number $m$ of constraints. Even though the running time of our algorithm depends on $m$, it is significantly faster for the important special case of totally unimodular linear programs, for which $1/δ\le n$ and which have only $O(n^2)$ constraints.