We shall revisit the entropy method for quasilinear Fokker-Planck equations with confinement to deduce exponential convergence to the equilibrium. Even for prototypical examples like the porous-medium equation, only one relative entropy has been known so far - the Ralston-Newman entropy, which is the analog of the logarithmic entropy in the linear case. We shall give a complete characterization of all admissible relative entropies for each quasilinear Fokker-Planck equation. In particular we find that fast-diffusion equations with power-law nonlinearities admit only one entropy, while porous medium equations give rise to a whole family of admissible relative entropies (similar to linear Fokker-Planck equations). These additional entropies then imply also new moment-control estimates on the porous-medium solution. Joint work with Jose Carrillo, Daniel Matthes.