I always believed the following statement: if $X$ is a smooth varieties over an algebraically closed field of positive characteristic. If we know that the general member of a base point free linear system $|L|$ is reduced, then indeed a general member is smooth.

However, I realize this is not obvious, though all the examples I know which fail this Bertini theorem has non-reduced fibers.

So I was wondering whether indeed this statement is true. Or counterexamples are known.