The involution width of finite simple groups

For a finite group generated by involutions, the involution width is defined to be the minimal $k\in\mathbb{N}$ such that any group element can be written as a product of at most $k$ involutions. We show that the involution width of every non-abelian finite simple group is at most $4$. This result is sharp, as there are families with involution width precisely 4.