A long series of recent results and breakthroughs have led to faster and better distributed approximation algorithms for single source shortest paths (SSSP) and related problems in the CONGEST model. The runtime of all these algorithms, however, is $\tildeΩ(\sqrt{n})$, regardless of the network topology, even on nice networks with a (poly)logarithmic network diameter $D$. While this is known to be necessary for some pathological networks, most topologies of interest are arguably not of this type. We give the first distributed approximation algorithms for shortest paths problems that adjust to the topology they are run on, thus achieving significantly faster running times on many topologies of interest. The running time of our algorithms depends on and is close to $Q$, where $Q$ is the quality of the best shortcut that exists for the given topology. While $Q = \tildeΘ(\sqrt{n} + D)$ for pathological worst-case topologies, many topologies of interest have $Q = \tildeΘ(D)$, which results in near instance optimal running times for our algorithm, given the trivial $Ω(D)$ lower bound. The problems we consider are as follows: (1) an approximate shortest path tree and SSSP distances, (2) a polylogarithmic size distance label for every node such that from the labels of any two nodes alone one can determine their distance (approximately), and (3) an (approximately) optimal flow for the transshipment problem. Our algorithms have a tunable tradeoff between running time and approximation ratio. Our fastest algorithms have an arbitrarily good polynomial approximation guarantee and an essentially optimal $\tilde{O}(Q)$ running time. On the other end of the spectrum, we achieve polylogarithmic approximations in $\tilde{O}(Q \cdot n^ε)$ rounds for any $ε> 0$.