We consider a quasi-linear parabolic (possibly, degenerate) equation with nonlinear dynamic boundary conditions. The corresponding class of initial and boundary value problems has already been studied previously, proving well-posedness of weak solutions and the existence of the global attractor, assuming that the nonlinearities are subcritical to a given exponent. The goal of this article is to show that the previous analysis can be redone for supercritical nonlinearities by proving an additional L∞-estimate on the solutions. In particular, we derive new conditions which reflect an exact balance between the internal and the boundary mechanisms involved, even when both the nonlinear sources contribute in opposite directions. Then, we show how to construct a trajectory attractor for the weak solutions of the associated parabolic system, and prove that any solution belonging to the attractor is bounded, which implies uniqueness. Finally, we also prove for the (semilinear) reaction–diffusion equation with nonlinear dynamic boundary conditions, that the fractal dimension of the global attractor is of the order ν−(N−1), as diffusion ν→0+, in any space dimension N⩾2, improving some recent results in Gal (2012) [23].