  In the period 1994-1999 Thas wrote a series of three papers on generalized
quadrangles of order $(s, s^2)$. In this Part IV we classify all finite
translation generalized quadrangles of order $(s, s^2)$ having a kernel of size
at least 3, containing a regular line not incident with the translation point.
There are several applications on generalized quadrangles of order $(s, s^2)$
having at least 2 translation points, on Moufang quadrangles, and concerning
the theorem of Fong and Seitz classifying all groups with a BN-pair of rank 2.
