Planar Turán number of the 7-cycle

The $\mathit{\text{planar}}\mathit{\text{Tur}}\acute{\mathit{\text{a}}}\mathit{\text{n}}\mathit{\text{number}}$ ${\text{ex}}_{\mathcal{P}}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_{\mathrm{\ell }}$ denote the cycle of length $\mathrm{\ell }$. The planar Turán number ${\text{ex}}_{\mathcal{P}}(n,C_{\mathrm{\ell }})$ behaves differently for $\mathrm{\ell }\le 10$ and for $\mathrm{\ell }\ge 11$, and it is known when $\mathrm{\ell }\in \{3,4,5,6\}$. We prove that ${\text{ex}}_{\mathcal{P}}(n,C_7)\le \frac{18n}{7}-\frac{48}{7}$ for all $n>38$, and show that equality holds for infinitely many integers $n$.