Tidal disruption events (TDEs) occur when a star is destroyed by a supermassive black hole at the centre of a galaxy, temporarily increasing the accretion rate on to the black hole and producing a bright flare across the electromagnetic spectrum. Radio observations of TDEs trace outflows and jets that may be produced. Radio detections of the outflows from TDEs are uncommon, with only about one-third of TDEs discovered to date having published radio detections. Here, we present over 2 yr of comprehensive, multiradio frequency monitoring observations of the TDE AT2019azh taken with the Very Large Array and MeerKAT radio telescopes from approximately 10 d pre-optical peak to 810 d post-optical peak. AT2019azh shows unusual radio emission for a thermal TDE, as it brightened very slowly over 2 yr, and showed fluctuations in the synchrotron energy index of the optically thin synchrotron emission from 450 d post-disruption. Based on the radio properties, we deduce that the outflow in this event is likely non-relativistic and could be explained by a spherical outflow arising from self-stream intersections or a mildly collimated outflow from accretion on to the supermassive black hole. This data set provides a significant contribution to the observational data base of outflows from TDEs, including the earliest radio detection of a non-relativistic TDE to date, relative to the optical discovery.