  We obtain a new bound on the number of two-rich points spanned by an
arrangement of low degree algebraic curves in $\mathbb{R}^4$. Specifically, we
show that an arrangement of $n$ algebraic curves determines at most $C_\epsilon
n^{4/3+3\epsilon}$ two-rich points, provided at most $n^{2/3+2\epsilon}$ curves
lie in any low degree hypersurface and at most $n^{1/3+\epsilon}$ curves lie in
any low degree surface. This result follows from a structure theorem about
arrangements of curves that determine many two-rich points.
