A two-dimensional MHD simulation of the Kelvin-Helmholtz (K-H) instability in a non-uniform density medium shows a strong development of turbulence through non-linear instabilities. The difference in density between the two media plays a crucial role on the fast turbulent mixing and transport. The onset of the turbulence is triggered not only by secondary K-H instability but also by Rayleigh-Taylor (R-T) instability at the density interface inside the normal K-H vortex, whose onset mechanisms are attributed to the centrifugal force of the rotating motion. The secondary R-T instability alters the macroscopic structure by transporting dense fluids to tenuous region, while the secondary K-H instability is just a seed for the turbulence. The onset mechanism and the formation of the broad mixing layer give a new understanding of the mixing process in a variety of geo- and astrophysical phenomena.