In 1979, Hakimi and Schmeichel  initiated such a study by determining the maximum number of triangles and 4-cycles possible in an $n$-vertex planar graph (see also  for a small correction).
 S. Hakimi, E. Schmeichel. On the Number of Cycles of Length k in a Maximal Planar Graph. J. Graph Theory 3 (1979): 69–86.
 A. Alameddine. On the Number of Cycles of Length 4 in a Maximal Planar Graph. J. GraphTheory, 4(1980): 417–422.
In 2019, Győri et al.  studied the maximum number of pentagons on planar graphs.
-  Győri E, Paulos A, Salia N, et al. The maximum number of pentagons in a planar graph[J]. arXiv preprint arXiv:1909.13532, 2019.
I have a problem about whether there is a similar result for planar bipartite graphs. For example, an upper bound on the number of 4 cycles (or any other even cycles). I haven't found it yet. If not, it is not clear what the difficulty is.
I once asked a similar question at Math. Stack and did not seem to see any results.