We present high resolution numerical simulations of magnetized plasma jets,modeled by means of the compressible magnetohydrodynamic equations. The computations employ Adaptive Mesh Refinement, which makes it possible to investigate long-term jet dynamics where both large-scale and small-scale effects are at play. We first discuss recent findings for periodic single shear flow layers at moderate Mach numbers (around unity) and large plasma beta values. In such cases, a trend to large scales occurs by continuous pairing/merging between adjacent vortices, simultaneously with the introduction of small-scale features by magnetic reconnection events. The vortices form as a result of Kelvin-Helmholtz unstable shear flow layers, and their coalescence arises from the growth of subharmonic modes at multiple wavelengths of the fastest growing Kelvin-Helmholtz instability. Extensions to 2D jets investigate how varying jet width alters the coalescence process occuring at both edges, e.g. by introducing bachelor coupling between vortices formed at opposing weakly magnetized, close shear layers. Finally, periodic segments of supersonic magnetized jets are simulated in two and three dimensional cases, which are characterized by violent shock-dominated transients.