  This work is a follow-up of the article [Proc.\ London Math.\ Soc.\
119(2):358--378, 2019], where the authors solved the problem of counting
labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic
estimate for the number $g_n$ of labelled 4-regular planar graphs on $n$
vertices. Our estimate is of the form $g_n \sim g\cdot n^{-7/2} \rho^{-n} n!$,
where $g>0$ is a constant and $\rho \approx 0.24377$ is the radius of
convergence of the generating function $\sum_{n\ge 0}g_n x^n/n!$, and conforms
to the universal pattern obtained previously in the enumeration of several
classes of planar graphs. In addition to analytic methods, our solution needs
intensive use of computer algebra in order to deal with large systems of
multivariate polynomial equations. We also obtain asymptotic estimates for the
number of 2- and 3-connected 4-regular planar graphs, and for the number of
4-regular simple maps, both connected and 2-connected.
