Villa's seven-game unbeaten run since Bruce took charge in October has lifted them to 11th in the Championship, just four points from the play-off zone.
"There'll be a few looking over their shoulders, which is good," said Bruce.
"But there is a big expectation. You have to be a big player and a good player to handle that expectation."
Following the win against Cardiff City, their third in a row at home, Bruce told BBC Sport: "We've done very little yet. There's a long way to go. But, quietly, with a few results, we're doing ok.
"This club's had a hammering over the last 18 months, but it's vitally important to see that bit of confidence coming through.
"We're still a 'work in progress' but there's been a huge improvement in seven weeks and we're now finding a way to win.
"With players like Albert Adomah, Jonathan Kodjia, Rudy Gestede and Jordan Ayew, we're showing a little bit of quality at the top end."
Bruce has told his players that if they keep producing the goods, there may be no need for him to strengthen in January, even though he has the green light to do so,
"If there's somebody out there who can make a difference I'm in a position where I can act," he added. "But I've told the players 'You're in the box seat'."
Bruce's men face another stern examination of their credentials over the next month as Villa face three of their next four games away from home.
After Saturday's visit to fifth-placed Leeds United, they then host Bruce's old club Wigan Athletic before two away trips in four days to another of his former clubs, Norwich City, then QPR.
Former Villa striker Chris Sutton has backed Bruce to get the club promoted this season.
"They've got momentum," the former Blackburn Rovers, Chelsea, Celtic and England striker, briefly one of Bruce's players at Birmingham City in 2006, prior to then joining Villa, told BBC Five Live's Monday night show.
"I think they will go up. He was a brilliant appointment. He's a brilliant manager. He's been around and proved himself. In fact, he should have been given a job at the top end of the end of the Premier League."
Aston Villa manager Steve Bruce says his players must deal with the growing expectation of becoming Championship promotion contenders.