Under the framework of dynamics on normal projective varieties by Kawamata, Nakayama and Zhang, and Hu and Li, we may reduce Kawaguchi–Silverman conjecture for automorphisms [Formula: see text] on normal projective threefolds [Formula: see text] with either the canonical divisor [Formula: see text] is trivial or negative Kodaira dimension to the following two cases: (i) [Formula: see text] is a primitively automorphism of a weak Calabi–Yau threefold, (ii) [Formula: see text] is a rationally connected threefold. And we prove Kawaguchi–Silverman conjecture is true for automorphisms of normal projective varieties [Formula: see text] with the irregularity [Formula: see text]. Finally, we discuss Kawaguchi–Silverman conjecture on normal projective varieties with Picard number two.