  The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a
graph $H$ is the maximum number of edges in an $n$-vertex planar graph without
$H$ as a subgraph. Let $C_k$ denote the cycle of length $k$. The planar Tur\'an
number $\textrm{ex}_{\mathcal P}(n,C_k)$ is known for $k\le 7$. We show that
dense planar graphs with a certain connectivity property (known as circuit
graphs) contain large near triangulations, and we use this result to obtain
consequences for planar Tur\'an numbers. In particular, we prove that there is
a constant $D$ so that $\textrm{ex}_{\mathcal P}(n,C_k) \le 3n - 6 -
Dn/k^{\log_2^3}$ for all $k, n\ge 4$. When $k \ge 11$ this bound is tight up to
the constant $D$ and proves a conjecture of Cranston, Lidick\'y, Liu, and
Shantanam.
