Dense $2$-connected planar graphs and the planar Turán number of $2C_k$

Shi, Walsh and Yu demonstrated that any dense planar graph with certain property (known as circuit graph) contains a large near-triangulation. We extend the result to $2$-connected plane graphs, thereby addressing a question posed by them. Using the result, we prove that the planar Tuán number of $2C_k$ is $[3-\mathrm{\Theta }(k^{{\log }_{2}3}{)}^{-1}]n$ when $k\ge 5$.