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The planar Tur\'an number ${\rm ex}_{\mathcal{P}}(n,C_k)$ is the largestnumber of edges in an $n$-vertex planar graph with no cycle of length $k$. Let$k\ge 11$ and $C,D$ be constants. Cranston, Lidick\'y, Liu and Shantanam\cite{2021Planar}, and independently Lan and Song \cite{LanSong} showed that${\rm ex}_{\mathcal{P}}(n,C_k)\ge 3n-6-\frac{Cn}{k}$ for large $n$. Moreover,Cranston et al. conjectured that ${\rm ex}_{\mathcal{P}}(n,C_k)\le3n-6-\frac{Dn}{k^{lg_23}}$ when $n$ is large. In this note, we prove that ${\rmex}_{\mathcal{P}}(n,C_k)\ge 3n-6-\frac{6\cdot 3^{lg_23}n}{k^{lg_23}}$ for every$k$. It implies Cranston et al.'s conjecture is essentially best possible.

A new construction for planar Tur\'an number of cycle