  In 2012 Monod introduced examples of groups of piecewise projective
homeomorphisms which are not amenable and which do not contain free subgroups,
and later Lodha and Moore introduced examples of finitely presented groups with
the same property. In this article we examine the normal subgroup structure of
these groups. Two important cases of our results are the groups $H$ and $G_0$.
We show that the group $H$ of piecewise projective homeomorphisms of
$\mathbb{R}$ has the property that $H"$ is simple and that every proper
quotient of $H$ is metabelian. We establish simplicity of the commutator
subgroup of the group $G_0$, which admits a presentation with $3$ generators
and $9$ relations. Further we show that every proper quotient of $G_0$ is
abelian. It follows that the normal subgroups of these groups are in bijective
correspondence with those of the abelian (or metabelian) quotient.
