This paper contains a method to prove the existence of smooth curves in positive characteristic whose Jacobians have unusual Newton polygon. Using this method, I give a new proof that there exist supersingular curves of genus $4$ for every prime $p$. More generally, for every prime $p$ and every $g \geq 4$, I prove that every Newton polygon whose $p$-rank is at least $g-4$ occurs for a smooth curve of genus $g$. In addition, I resolve some cases of Oort's conjecture about Newton polygons.
