The planar Turán number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Turán number of $k(k\geq 3)$ vertex-disjoint union of cycles is the trivial value $3n-6$. We determine the planar Turán number of $C_{3}\text{-}C_{3}$ and $C_{3}\text{-}C_{4}$, where $C_{k}\text{-}C_{\ell}$ denotes the graph consisting of two disjoint cycles $C_k$ with an edge connecting them.