  Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$-sphere $S^2$ induces
a pull-back operation on Jordan curves $\alpha\subset S^2\setminus P_f$, where
$P_f$ is the postcritical set of $f$. Here the isotopy class $[f^{-1}(\alpha)]$
(relative to $P_f$) only depends on the isotopy class $[\alpha]$. We study this
operation for Thurston maps with four postcritical points. In this case a
Thurston obstruction for the map $f$ can be seen as a fixed point of the
pull-back operation.
  We show that if a Thurston map $f$ with a hyperbolic orbifold and four
postcritical points has a Thurston obstruction, then one can "blow up" suitable
arcs in the underlying $2$-sphere and construct a new Thurston map $\widehat f$
for which this obstruction is eliminated. We prove that no other obstruction
arises and so $\widehat f$ is realized by a rational map. In particular, this
allows for the combinatorial construction of a large class of rational Thurston
maps with four postcritical points.
  We also study the dynamics of the pull-back operation under iteration. We
exhibit a subclass of our rational Thurston maps with four postcritical points
for which we can give positive answer to the global curve attractor problem.
