The Premier League clash ended in a goalless draw.
Sunderland are currently just three points above the Premier League relegation zone.
“I think we have had five or six draws from seven games and we need a win to build confidence,” stated Bruce.
“You can see there is nervousness in the team and we need to cope with that.
“We have to stick together and try and gain some confidence from somewhere, and that can only come with a win.
“We surrendered possession, we were too direct and we couldn’t find the final pass. It was very frustrating, but Fulham played well.”
The former Wigan boss added: “Listen, I am the manager, the buck stops with me. I have brought five or six of them to the club and anything that’s levelled at any of them, it is my job to try to protect them if I can, which I will do.
“Of course it’s my responsibility and it hasn’t been easy, that’s for sure. But the one thing I am not going to do is shy away from it.
“I always knew it was going to be difficult. The great start we got off to maybe led us up a path we weren’t ready for.
“However, I didn’t want this to happen, that’s for sure, and the only thing that’s left for me to do is make sure I stick with it.
“I think I have been decent at the job in the past and I am convinced I am still right to do it.”