I am looking for "natural" examples of a finitely generated group which have uncountably many normal subgroups but are not SQ-universal. A group $G$ is SQ-universal if for any countable group $H$, $H$ embeds into some quotient of $G$. For example, non-elementary hyperbolic groups are known to be SQ-universal (in fact acylindrically hyperbolic groups are SQ-universal)
For the most part, by natural I just mean it is not difficult to prove that the group exists. A non example would be a Burnside group of sufficiently high exponent. Ideally it would be a reasonably well known group, maybe even a linear group or something like that.
To be SQ-universal, the group must contain a free group of rank 2, so maybe looking at groups which don't contain such free groups would be a good place to look.
(In writing this question I found an acceptable group, so answered, but I welcome more examples)