We introduce a new class of Green–Naghdi type models for the propagation of internal waves between two (1+1)-dimensional layers of homogeneous,immiscible, ideal, incompressible, and irrotational fluids, vertically delimited by a flat bottom and a rigid lid. These models are tailored to improve the frequency dispersion of the original bi-layer Green–Naghdi model, and in particular to manage high-frequency Kelvin–Helmholtz instabilities, while maintaining its precision in the sense of consistency. Our models preserve the Hamiltonian structure, symmetry groups, and conserved quantities of the original model. We provide a rigorous justification of a class of our models thanks to consistency, well-posedness, and stability results. These results apply in particular to the original Green–Naghdi model as well as to the Saint–Venant (hydrostatic shallow water) system with surface tension.