The planar Turán number of a graph H, denoted by exP(n,H)\documentclass[12pt]{minimal}
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\begin{document}$$ex_{_\mathcal {P}}(n,H)$$\end{document}, is the maximum number of edges in a planar graph on n vertices without containing H as a subgraph. This notion was introduced by Dowden in 2016 and has attracted quite some attention since then; those work mainly focus on finding exP(n,H)\documentclass[12pt]{minimal}
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\begin{document}$$ex_{_\mathcal {P}}(n,H)$$\end{document} when H is a cycle or Theta graph or H has maximum degree at least four. In this paper, we first completely determine the exact values of exP(n,H)\documentclass[12pt]{minimal}
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\begin{document}$$ex_{_\mathcal {P}}(n,H)$$\end{document} when H is a cubic graph. We then prove that exP(n,2C3)=⌈5n/2⌉-5\documentclass[12pt]{minimal}
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\begin{document}$$ex_{_\mathcal {P}}(n,2C_3)=\lceil 5n/2\rceil -5$$\end{document} for all n≥6\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 6$$\end{document}, and obtain the lower bounds of exP(n,2Ck)\documentclass[12pt]{minimal}
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\begin{document}$$ex_{_\mathcal {P}}(n,2C_k)$$\end{document} for all n≥2k≥8\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2k\ge 8$$\end{document}. Finally, we also completely determine the exact values of exP(n,K2,t)\documentclass[12pt]{minimal}
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\begin{document}$$ex_{_\mathcal {P}}(n,K_{2,t})$$\end{document} for all t≥3\documentclass[12pt]{minimal}
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\begin{document}$$t\ge 3$$\end{document} and n≥t+2\documentclass[12pt]{minimal}
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\begin{document}$$n\ge t+2$$\end{document}.
