  Let $\mathcal{H}$ be a set of graphs. The planar Tur\'an number,
$ex_\mathcal{P}(n,\mathcal{H})$, is the maximum number of edges in an
$n$-vertex planar graph which does not contain any member of $\mathcal{H}$ as a
subgraph. When $\mathcal{H}=\{H\}$ has only one element, we usually write
$ex_\mathcal{P}(n,H)$ instead. The topic of extremal planar graphs was
initiated by Dowden (2016). He obtained sharp upper bound for both
$ex_\mathcal{P}(n,C_5)$ and $ex_\mathcal{P}(n,K_4)$. Later on, we obtained
sharper bound for $ex_\mathcal{P}(n,\{K_4,C_7\})$. In this paper, we give upper
bounds of $ex_\mathcal{P}(n,\{K_4,C_5\})\leq {15\over 7}(n-2)$ and
$ex_\mathcal{P}(n,\{K_4,C_6\})\leq {7\over 3}(n-2)$. We also give constructions
which show the bounds are tight for infinitely many graphs.
