Premier League Preview: Crystal Palace v Hull
Bruce's side head to Selhurst Park on the back of a torrid run of form, with just two points from their last six outings.
That sequence has seen them fall to 17th, above a resurgent Leicester City by a solitary goal.
As well as clashes with Palace and fellow strugglers Burnley, Hull's run-in includes fixtures against four sides currently in the top six - Arsenal, Manchester United, Liverpool and Tottenham.
"One or two clubs have moved themselves away from it in the last two or three weeks," Bruce, who had a brief stint at Palace in 2001, told talkSPORT.
"A month ago, there were probably eight teams in trouble but if we're being honest it now looks as though there are five of us. Obviously we now aim to finish above at least three of those.
"Of our six remaining games, four are at home. There's no question that it's going to be difficult. However, the way I see it is that we need to win at least two games.
"We're going to Crystal Palace first up on Saturday and of course they've had a great run of late, but we know we're capable.
"It doesn't really matter if you're playing a side at the top, a side in mid-table or a side down at the bottom, they're all difficult games.
"We just hope that we can perform and get the performance levels up to the degree where we give ourselves a fair chance.
"We're realistic enough to know that we've got a difficult challenge, but we also know that we've got as good a chance as anyone of coming out of it right."
Since Alan Pardew's arrival, Palace have turned themselves from relegation battlers to candidates for a top-half finish, although a run of four straight victories ended with a surprise 2-0 home reverse to West Brom last time out.
"I think it [a top-10 finish] is something at the end of the season to look at, not now," Pardew said. "My focus is on the game.
"I do want a top-10 finish because we got ourselves in that position and that certainly wasn't the case when I walked in. It's something we've done well.
"I'm proud of the players and their achievement. And the fans, we owe them a little bit, because we let them down [last week]."
Pardew is still without Marouane Chamakh and former Hull man Fraizer Campbell (both hamstring), with Jordon Mutch (thigh) and Jerome Thomas (knock) remaining doubts.
For the visitors, David Meyler serves the last instalment of a three-match ban, while Curtis Davies (thigh) and Gaston Ramirez (hamstring) are still fitness concerns for Bruce.