The minimum number of vertices one can remove in order to make the graph acyclic (which is the only meaning I can give to "one outer face") is called the *decycling number* of a graph. There is a famous conjecture here by Albertson and Berman, and independently Akiyama and Watanabe which says that the decycling number of a planar graph on $n$ vertices is at most $n/2$. This appears to be a very difficult problem. It is known to be true for triangle free planar graphs.

In fact for triangle free planar graphs (and therefore bipartite planar graphs) the decycling number is $\le \frac{15n-24}{32}$. This is proved in "Large Induced Forests in Triangle-free Planar Graphs" by M.R. Salavatipour. It was recently improved to $\frac{57n-72}{128}$ by L. Kowalik, B. Lužar, R. Škrekovski in "An improved bound on the largest induced forests for triangle-free planar graphs". The proofs make use of the Discharging method, which you might have seen in the context of the 4 color theorem.

For the class of non-planar bipartite graphs, the only results I've seen are for sparse bipartite graphs. See section 2 in the recent article "Short proofs of some extremal results" by Conlon, Fox, and Sudakov.