We investigate the dynamical spreading of correlations in many-body quantum systems with competing short- and global-range couplings. We monitor the nonequilibrium dynamics of the correlations following a quench, showing that for strong short-range couplings the propagation of correlations is dominated at short and intermediate distances by a causal, light-cone dynamics, resembling the purely short-range quantum systems. However, the interplay of short- and global-range couplings leads to a crossover between space-time regions in which the light cone persists to regions where a supersonic, distance-independent spreading of the correlations occurs. We identify the important ingredients needed for capturing the supersonic spreading and demonstrate our findings in systems of interacting bosonic atoms, in which the global-range coupling is realized by a coupling to a cavity light field or atomic long-range interactions, respectively. We show that our results hold in both one and two dimensions and in the presence of dissipation. Furthermore, we characterize the short-time power-law scaling of the distance-independent growth of the density-density correlations.