Johnson, who left League One rivals Yeovil at the end of September to replace Brian Tinnion, has won just three of his 11 matches in charge as the Robins’ league position has gradually worsened under his tenure.
However, Sexstone insists the former Latvia coach still has the backing of the club's board.
"Gary has a long-term contract at the club and we have every confidence he will be successful, otherwise we wouldn't have given him the job," Sexstone told PA Sport.
"Of course he will be given time to turn things around, it is not even an issue for discussion.
"We have every faith in Gary and he has our full backing. I am actually rather reluctant to talk about it as it draws attention to an issue that is really not an issue at this football club."