Ferromagnetic nanowires are likely to play an important role in future spintronic devices. Magnetic domain walls, which separate regions of opposing magnetization in a nanowire, can be manipulated and used to encode information for storage or to perform logic operations. Owing to their reduced size and dimensionality, the characterization of domain-wall motion is an important problem. To compete with other technologies, high-speed operation, and hence fast wall propagation, is essential. However, the domain-wall dynamics in nanowires has only been investigated in the last five years and some results indicate a drastic slowing down of wall motion in higher magnetic fields. Here we show that the velocity-field characteristic of a domain wall in a nanowire shows two linear regimes, with the wall mobility at high fields reduced tenfold from that at low fields. The transition is marked by a region of negative differential mobility and highly irregular wall motion. These results are in accord with theoretical predictions that, above a threshold field, uniform wall movement gives way to turbulent wall motion, leading to a substantial drop in wall mobility. Our results help resolve contradictory reports of wall propagation velocities in laterally confined geometries, and underscore the importance of understanding and enhancing the breakdown field for practical applications.