  Given a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le
14$, we study projections $S\hookrightarrow \mathbb P^g\dashrightarrow \mathbb
P^2$ of minimal degree and their variational structure. In particular, we prove
that the degree of irrationality of all such surfaces is at most $4$, and that
for $g=7,8,9,11$ there are no rational maps $S\dashrightarrow \mathbb P^2$ of
degree $3$ induced by the primitive linear system. Our methods combine vector
bundle techniques \`a la Lazarsfeld with derived category tools, and also make
use of the rich theory of singular curves on $K3$ surfaces.
