Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\to \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{N}$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.