Comment: The presented decomposition technique (Theorems 1.2/1.3) has been already independently shown by T. Ueckerdt, D.R. Wood, W. Yi (https://doi.org/10.37236/10614); a circumstance that I missed due to the result not being advertised in the corresponding abstract. Moreover, Lemma 4.2 is wrong, thus new technical details are necessary. I would like to thank Sergey Pupyrev for pointing this out

As an alternative to previously existing planar graph product structure theorems, we prove that every planar graph $G$ is a subgraph of the strong product of $K_2$, a path and a planar subgraph of a $4$-tree. As an application, we show that the queue number of planar graphs is at most $38$ whereas the queue number of planar bipartite graphs is at most $25$.