Unipotent elements of nonprime order in representations of the classical algebraic groups: two big Jordan blocks

For irreducible rational representations of the classical algebraic groups in characteristic $p>2$ that are not equivalent to a composition of a group morphism and the standard representation, it is proved that usually the image of a unipotent element of order $p^{s+1}>p$ has at least two Jordan blocks of size $>p^s$; all exceptions are indicated explicitly. As a corollary, irreducible rational representations of these groups whose images contain unipotent elements with just one Jordan block of size $>1$ are classified.