Johnson, who joined Tottenham from Nottingham Forest in a £47.5million deal in the summer, was among those guiltiest of squandering promising positions on Thursday night as Spurs were beaten by West Ham.
Johnson has scored just once in a Tottenham shirt and Postecoglou believes the 22-year-old needs more self-belief.
“He’s joined a big football club," Postecoglou said. "You can’t play for a big football club and hide away. It doesn’t happen.
“You look at any player who goes to a big football club, that’s where they want to be, they want the responsibility, they want the limelight, they want to be the person who makes that difference.
“That’s why Brennan came here. He could have very easily stayed at Nottingham Forest where he was making a great career for himself. But he wanted to get out of his comfort zone because he had been there for so long and he chose to come to a big club.
"Well, that’s part of being at a big club. I know he knows that. But that’s not just an overnight thing. He’ll come to embrace that. We’ve seen elements of his game which are very, very exciting and we’ve got to get him to do that more and believe in himself more.
“I know that will come. I have full faith that it will come. I’ve seen that in him. But like any other process with young players and young people, they have got to get to that space. You try to help them to get there as quickly as possible but sometimes it takes a little bit longer."
Johnson could face competition from Richarlison for a place in the XI against Newcastle, although Sunday may come too soon for the Brazilian to start after he had surgery on a persistent groin problem last month.
Postecoglou says the operation was a success and challenged Richarlison to make an impact in the side.
"Fair to say he looks a lot freer and better physically than he did earlier in the year when he was struggling," the Australian said. “It seems like the medical intervention has been positive for him.
"He’s short of some match fitness but it was good to get him on last night for a bit more time.
“We’re going to need him, we’re going to need all the guys. The group that we’ve got at the moment is what we’re going to have between now and the foreseeable future. They’re the ones that are going to have turn their fortunes around."