We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in Aleksandrov, Peller, Potapov and Sukochev ['Functions of normal operators under perturbations', Adv. Math. 226 (2011) 5216-5251.]. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators f(N ₁)R-Rf(N ₂) in terms of N ₁R-RN ₂, where N ₁ and N ₂ are normal operator and R is a bounded linear operator. In particular, we show that if 0 < α < 1 and f is a Hölder function of order α, then, for normal operators N ₁ and N ₂, In the last section, we obtain lower estimates for constants in operator Hölder estimates.