We consider the transverse structure and stability properties of relativistic jets formed in the course of the collapse of a massive progenitor. Our numerical simulations show the presence of a strong shear in the bulk velocity of such jets. This shear can be responsible for a very rapid shear–driven instability that arises for any velocity profile. This conclusion has been confirmed both by numerical simulations and theoretical analysis. The instability leads to rapid fluctuations of the main hydrodynamical parameters (density, pressure, Lorentz factor, etc.). However, the perturbations of the density are e ectively decoupled from those of the pressure because the beam of the jet is radiation–dominated. The characteristic growth time of instability is much shorter than the life time of the jet and, therefore, may lead to a complete turbulent beam. In the course of the non-linear evolution, these fluctuations may yield to internal shocks which can be randomly distributed in the jet. In the case that internal shocks in a ultrarelativistic outflow are responsible for the observed phenomenology of gamma-ray bursts, the proposed instability can well account for the short-term variability of gamma-ray light curves down to milliseconds.