The star chromatic index of a graph $G$, denoted by $\chi {{}^{\prime }}_{st}(G)$, is the smallest number of colors required to properly color $E(G)$ such that every connected bicolored subgraph is a path with no more than three edges. A graph is $K_{2,t}$-free if it contains no $K_{2,t}$ as a subgraph. This paper proves that every $K_{2,t}$-free planar graph $G$ satisfies ${\chi }_{st}^{\prime }(G)\le 1.5\mathrm{\Delta }+20t+20$, which is sharp up to the constant term. In particular, our result provides a common generalization of previous results on star edge coloring of outerplanar graphs by Bezegová et al.(2016) and of $C_4$-free planar graphs by Wang et al.(2018), as those graphs are subclasses of $K_{2,3}$-free planar graphs.