In this study, a novel multilayer terahertz metamaterial with four graphene sub-structures is designed to achieve dynamically tunable triple plasmon-induced transparency (PIT) due to interference among bright-bright modes. Coupled mode theory (CMT) and finite-difference time-domain (FDTD) simulations show a high degree of consistency in their results. Interestingly, enhanced transmission dips are observed when various graphene structures couple with the two longitudinal graphene strips at the bottom layer. Based on the dynamic modulation characteristics of graphene, the metamaterial demonstrates seven-band optical switching capabilities, achieving modulation depths (MD) of 94.6%, 88.1%, 90.8%, 90.8%, 90.4%, 86.6%, and 87.5%. Additionally, the proposed graphene metamaterial also shows excellent slow-light effects, with a group index reaching up to 1034. This proposed terahertz metamaterial holds significant theoretical implications for the development of dynamically integrated terahertz optoelectronic devices.