Fractional optical vortices in the terahertz (THz) regime are supposed to have unique applications in various areas, i.e., THz communications, optical manipulations, and THz imaging. However, it is still challenging to generate and manipulate high-power THz vortices. Here, we present a way to generate intense THz vortex beams with a continuously tunable topological charge by injecting a weakly relativistic ultrashort laser pulse into a parabolic plasma channel. By adjusting the injection conditions of the laser pulse, the trajectory of the laser centroid can be twisted into a cylindrical spiral, along which laser wakefields are also excited. Due to the inhomogeneous transverse density profile of the plasma channel and laser wakefield excitation, intense THz radiation carrying orbital angular momentum is produced with field strength reaching sub-GV/m, even though the drive laser energy is at a few tens of mJ. The topological charge of such a radiation is determined by the laser trajectories, which are continuously tunable as demonstrated by theoretical analysis as well as three-dimensional particle-in-cell simulations. Such THz vortices with unique properties may find applications in broad areas.