We report the topology-mediated modulation of a twisted domain wall speed in a thick perpendicularly magnetized system. By exploiting the topological robustness of the direction of the Bloch wall component in the twisted domain wall, we show that the domain wall speed either increases or decreases depending on whether the transverse magnetic field is parallel or antiparallel to the Bloch wall component. The decrease in the speed is maintained until the antiparallel transverse reaches ∼0.3 T, indicating that the twisted domain wall can offer wide controllability supported by the topological robustness which involves an injection of a Bloch point. We also demonstrate that the transverse magnetic field suppresses the Walker breakdown, allowing high mobility domain wall motion for a wide range of perpendicular driving fields.