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Abstract Every Thurston map$$f\colon S^2\rightarrow S^2$$on a$$2$$-sphere$$S^2$$induces a pull-back operation on Jordan curves$$\alpha \subset S^2\smallsetminus {P_f}$$, where$${P_f}$$is the postcritical set off. 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 mapfcan be seen as a fixed point of the pull-back operation. We show that if a Thurston mapfwith 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.