Translation:He has not walked since he was sick.
"Se tornou doente" is very strong. It would mean he became sick in essence, permanently. You wouldn't say that.
But "ficou doente" is transitory, and the good option.
You could compare "se tornou" with "ser", and "ficou" with "estar" in this case.
Just don't say "se tornou doente". That's bad.