Bradley's side enjoyed a morale-boosting 3-0 win over fellow strugglers Sunderland the previous weekend but consecutive defeats to West Bromwich Albion and Boro have raised their relegation fears once more.
Swansea next have consecutive home games against West Ham United and Bournemouth, matches Bradley said would be crucial in their battle to beat the drop.
"When I came here I knew what I was getting into and I'm not backing down from it now," Bradley, who became the first American to coach a Premier League side when he replaced Francesco Guidolin in October, told the BBC.
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"That's the challenge. When the team has put itself in a tough spot, everybody has got to stick together during the toughest times and fight the fight.
"I put pressure on myself to see if we can get stronger and fight for points. I will continue to do that."
Swansea have conceded 15 goals in their last five matches which included an extraordinary 5-4 victory over Crystal Palace -- Bradley's first win in charge of the team.
"It's the same story. We start OK, but concede. Before we know it we're 2-0 down," he said. "I feel as though we put ourselves in a hole over and over and that takes a lot of out of a team."
Swansea are still only three points behind fourth-from-bottom Palace and Bradley is confident his side can climb away from trouble if they beat the sides around them.
"Up until today we have done well in the games against the teams who are right around us in the table," Bradley said.
"We talked about the importance of the games against Crystal Palace and Sunderland, but for the first time here we didn't take care of business (against a direct rival).
"That's going to be important, but now we have to look to our next two matches and try to win those. They are going to be key for us."
(Reporting by Martyn Herman; Editing by John O'Brien)
