  In this paper we estimate the planar Tur\'an number
$\mathrm{ex}_\mathcal{P}(n,H)$ of some graphs $H$, i.e., the maximum number of
edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph.
We give a new, short proof when $H=C_5$, and study the cases when $G$ is
bipartite or triangle-free and $H$ is a short even cycle. The proofs are mostly
new applications or variants of the "contribution method" introduced by Ghosh,
Gy\H{o}ri, Martin, Paulos and Xiao in arXiv:2004.14094.
