Recent reports of current-induced switching of ferrimagnetic oxides coupled to heavy metals have opened prospects for implementing magnetic insulators into electrically addressable devices. However, the configuration and dynamics of magnetic domain walls driven by electrical currents in insulating oxides remain unexplored. Here we investigate the internal structure of the domain walls in Tm3Fe5O12 (TmIG) and TmIG/Pt bilayers, and demonstrate their efficient manipulation by spin-orbit torques with velocities of up to 400 ms-1 and minimal current threshold for domain wall flow of 5 × 106 A cm-2. Domain wall racetracks are defined by Pt current lines on continuous TmIG films, which allows for patterning the magnetic landscape of TmIG in a fast and reversible way. Scanning nitrogen-vacancy magnetometry reveals that the domain walls of TmIG thin films grown on Gd3Sc2Ga3O12 exhibit left-handed Néel chirality, changing to an intermediate Néel-Bloch configuration upon Pt deposition. These results indicate the presence of interfacial Dzyaloshinskii-Moriya interaction in magnetic garnets, opening the possibility to stabilize chiral spin textures in centrosymmetric magnetic insulators.