Recognition by spectrum for simple classical groups in characteristic $2$

A finite group $G$ is said to be recognizable by spectrum if every finite group with the same set of element orders as $G$ is isomorphic to $G$. We prove that all finite simple symplectic and orthogonal groups over fields of characteristic $2$, except $S_4(q)$, $S_6(2)$, $O^+_8(2)$ and $S_8(q)$, are recognizable by spectrum. This result completes the study of the recognition-by-spectrum problem for finite simple classical groups in characteristic $2$.