The main result of this paper is the resolvent similarity criterion which says that linear growth of the resolvent towards the spectrum is sufficient for a Hilbert space contraction with finite rank defect operators and spectrum not covering the unit disc to be similar to a normal operator. Similar results are proved for operators having a spectral set bounded by a Dini-smooth Jordan curve; in particular, a dissipative operator with finite rank imaginary part is similar to a normal operator if and only if its resolvent grows linearly towards the spectrum. Relevant results on the insufficiency of linear resolvent growth not accompanied by smallness of defect operators are presented. Also it is proved that there is no restriction on the spectrum, other than finiteness, which together with linear resolvent growth implies similarity to a normal operator. The construction of corresponding examples depends on a characterization of well-known Ahlfors curves as curves of linear length growth with respect to linear fractional transformations.