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 \}.
