I have been reading through a book of Robinson where it is mentioned (informally!) that solvable groups have "many" subnormal subgroups (subgroups $H<G$ with $H=H_0 \lhd H_1 \lhd \ldots \lhd H_n = G$). There seems to be quite a body of work on this, but it brought me to ponder on:

$\textbf{Question:}$ Given an infinite solvable group $G$, are there "few" malnormal subgroups (i.e. $H \cap gHg^{-1} = e$ for any $g \in G \setminus H$)?

I'm open to any interpretation of "few" (from "none" to "if there is one then it must be [insert property]" to ...) or any example.

[Edit: I forgot to put "finitely generated" in the question.]