In this paper we address the Cauchy problem for two systems modeling the propagation of long gravity waves in a layer of homogeneous, incompressible and inviscid fluid delimited above by a free surface, and below by a non-necessarily flat rigid bottom. Concerning the Green–Naghdi system, we improve the result of Alvarez–Samaniego and Lannes (Indiana Univ. Math. J., 57(1):97–131, 2008) in the sense that much less regular data are allowed, and no loss of derivatives is involved. Concerning the Boussinesq–Peregrine system, we improve the lower bound on the time of existence provided by Mésognon-Gireau (Adv. Differ. Equ., 22(7-8):457–504, 2017). The main ingredient is a physically motivated change of unknowns revealing the quasilinear structure of the systems, from which energy methods are implemented.