A defining feature of topologically ordered states of matter is the existence of locally indistinguishable states on spaces with nontrivial topology. These degenerate states form a representation of the mapping class group (MCG) of the space, which is generated by braids of defects or anyons, and by Dehn twists along noncontractible cycles. These operations can be viewed as fault-tolerant logical gates in the context of topological quantum error correcting codes and topological quantum computation. Here, we show that braids and Dehn twists can in general be implemented by a constant depth quantum circuit, with a depth that is independent of code distance d and system size. The circuit consists of a constant depth local quantum circuit (LQC) implementing a local geometry deformation of the quantum state, followed by a permutation on (relabeling of) the qubits. The permutation requires permuting qubits that are separated by a distance of order d; it can be implemented by collective classical motion of mobile qubits or as a constant depth circuit using long-range SWAP operations (with a range set by d) on immobile qubits. We further show that (i) applying a given braid or Dehn twist k times can be achieved with O(log k) time overhead, independent of code distance and system size, which implies an exponential speedup for certain logical gate sequences by trading space for time, and (ii) an arbitrary element of the MCG can be implemented by a constant depth (independent of d) LQC followed by a permutation, where in this case the range of interactions of the LQC grows with the number of generators in the presentation of the group element. Applying these results to certain non-Abelian quantum error correcting codes demonstrates how universal logical gate sets can be implemented on encoded qubits using only constant depth unitary circuits.