  We prove that every postsingularly finite entire map $g$ can be approximated
by a sequence of postcritically finite complex polynomials $(g_n)$ such that
their postsingular dynamics $g|P_g$ and $g_n|P_{g_n}$ are conjugate for every
$n \in \mathbb{N}$. To establish this result, we introduce the notion of
combinatorial convergence for sequences of entire Thurston maps defined on the
topological plane $\mathbb{R}^2$ and having the same marked set $A$. We prove
that if such a sequence $(f_n)$ converges combinatorially to a Thurston map
$f$, then the sequence of Thurston pullback maps $(\sigma_{f_n})$ converges to
$\sigma_f$ locally uniformly on the Teichm\"{u}ller space
$\mathrm{Teich}(\mathbb{R}^2, A)$.
