  We study the weight 2 graded piece of the compactly supported rational
cohomology of the moduli spaces of curves $M_{g,n}$ and show that this can be
computed as the cohomology of a graph complex that is closely related to graph
complexes arising in the study of embedding spaces. For $n = 0$, we express
this cohomology in terms of the weight zero compactly supported cohomology of
$M_{g',n'}$ for $g' \leq g$ and $n' \leq 2$, and thereby produce several new
infinite families of nonvanishing unstable cohomology groups on $M_g$,
including the first such families in odd degrees. In particular, we show that
the dimension of $H^{4g-k}(M_g)$ grows at least exponentially with $g$, for $k
\in \{ 8, 9, 11, 12, 14, 15, 16, 18, 19 \}$.
