Bradley, the first American manager in the league, began his tenure with a 3-2 defeat away to Arsenal last weekend.
That left Swansea bottom but one, having taken only one point from their last seven games.
In five seasons in the Premier League they have never finished outside the top 12 but now face a relegation fight.
"It was a big topic in my first year in Norway (with Stabaek), where the goal was to stay up after promotion," Bradley told a news conference.
"Every expert said we were going down.
"We started well but them had a dip and the press said 'here it comes'. I told the team not to listen to them because they don’t watch us train or see what we’re about."
The former U.S. national team manager eventually took the Norwegian club into the Europa League before leaving to take over at Le Havre in France.
"The situations are similar but the spotlight is bigger here," he said. "When you have the history of a club like this, what it means for the community, I understand how important it is to make us better."
Mid-table Watford provide the opposition on Saturday in Bradley's first home match and he said results at the Liberty Stadium will be key to moving up the table.
"There are no free points, we have to fight every week and playing at the Liberty has to be a big focus. We have talked about needing to do well at home and Saturday we have a chance."
But they must cut out the mistakes that led to the narrow defeat to Arsenal, he added.
"It’s about being able to do two things - can we continue to improve ourselves and make our football, mentality and intensity better and eliminate the mistakes?
"We have to be very positive and view this as a very important match for the supporters and for confidence."
(Reporting by Steve Tongue, editing by Pritha Sarkar)
