This work studied the slow dynamics of liquids with mesoscopic structure and its relation to shear viscosity. Quasielastic scattering measurements were made on a liquid higher alcohol, 3,7-dimethyl-1-octanol, using γ-ray time-domain interferometry at a synchrotron radiation facility, SPring-8. The quasielastic scattering spectra were measured to determine the structural relaxation at two wavenumbers of the prepeak and the main peak of the static structure factor. It was found that relaxation at the prepeak is more than 10 times slower than that at the main peak. Compared with the viscoelastic spectrum, which exhibits bimodal relaxation, the relaxations at the prepeak and the main peak were shown to correspond to the slower and faster modes of the viscoelastic relaxation, respectively. This indicates that the dynamics of the mesoscopic structure represented as the prepeak contributes to the shear viscosity through the slowest mode of the viscoelastic relaxation.