We show experimentally that in a supercooled liquid composed of molecules with internal degrees of freedom the internal modes contribute to the frequency-dependent shear viscosity and damping of transverse phonons, which results in an additional broadening of the transverse Brillouin lines. Earlier, only the effect of internal modes on the frequency-dependent bulk viscosity and damping of longitudinal phonons was observed and explained theoretically in the limit of weak coupling of internal degrees of freedom to translational motion. A new theory is needed to describe this new effect. We also demonstrate that the contributions of structural relaxation and internal processes to the width of the Brillouin lines can be separated by measurements under high pressure.