For a long time, it has been well recognized that there exists a deep link between the fast vibrational excitations and the slow diffusive dynamics in glass-forming systems. However, it remains as an open question whether and how the short-time scale dynamics associated with vibrational intrabasin excitations is related to the long-time dynamics associated with diffusive interbasin hoppings. In this paper we briefly review the research progress that addresses this challenge. By identifying a structural order parameter—local connectivity of a particle which is defined as the number of nearest neighbors having the same local spatial symmetry, it is found that the local connectivity can tune and modulate both the short-time vibrational dynamics and the long-time relaxation dynamics of the studied particles in a model of metallic supercooled liquid. Furthermore, it reveals that the local connectivity leads the long-time decay of the correlation functions to change from stretched exponentials to compressed ones, indicating a dynamic crossover from diffusive to hyperdiffusive motions. This is the first time to report that in supercooled liquids the particles with particular spatial symmetry can present a faster-than-exponential relaxation that has so far only been reported in out-of-equilibrium materials. The recent results suggest a structural bridge to link the fast vibrational dynamics to the slow structural relaxation in glass-forming systems and extends the compressed exponential relaxation phenomenon from earlier reported out-of-equilibrium materials to the metastable supercooled liquids.