Jordon Mutch's stoppage-time strike earned Cardiff a first away win since their top-flight return and there were jeers directed at Jol at the final whistle.
Fulham have one point from their last seven home matches in the Barclays Premier League and their next opportunity to win at Craven Cottage will come against Stoke - managed by Jol's predecessor Mark Hughes - next Saturday, when six months will have elapsed since their last home success in the league, the April 1 defeat of QPR.
"I'll always be calm, but I hate it when people are getting worried and nervous, especially when they ask a manager to quit," Jol said.
"That is awful, because you never know what will happen with a new manager.
"If we are bottom of the table at the end of the season it's not about me, because I won't be here, it's about the club and I don't want to see this club at the bottom of the table.
"They (the fans) expect us to do better.
"If you are in the bottom three or bottom five, even if it's early in the season, you get pressure.
"I take responsibility, because I can't say that the players don't work hard."
He added: "Fulham is not a club who should be in the top six. We were 20th in the league in spending money."
Asked if he has the confidence of Fulham owner Shahid Khan, the billionaire owner of the Jacksonville Jaguars NFL franchise, Jol pointed to the key difference with American sport.
Jol said: "For the owner it's different. In America you can't go down. (If) the Jaguars are bottom of the table, they can't go down.
"In England you can go down, so that makes people nervous. I can appreciate that."
Asked if he is now nervous about his own position, Jol said: "I know that we can do a lot better.
"I've got players who have played 300 games. We had all the size in the world today and we still lost headers in the far post, even when we are 6ft 6ins or 7ins. And that is worrying."
After conceding from a last-gasp set-piece to suffer a draw with West Brom in the previous Premier League contest at Craven Cottage, Fulham left Steven Caulker unmarked to head in Cardiff's opener and desperate defending played a part in Mutch's winner, well struck though it was. Bryan Ruiz had equalised before the interval.
"You have to defend, you have to defend especially corner kicks," Jol said.
"That is the easiest way to defend. The only thing you have to do is do something against your opponent.
"You talk about it, you do it in training and the next game they score from a corner kick. It's awful."
Jol conceded Cardiff were the better side and his counterpart Malky Mackay described his side's performance as dominant.
The City boss said: "I was delighted with the way we played today, from the first minute till the last. We ended up with 24 attempts at goal and away from home that's a fantastic stat.
"We kept striving for the win. We put on three substitutions to try to win the game and one of the players ended up scoring a fantastic goal.
"I think we got our just rewards in the end. We deserved to win the game."
Mackay praised Chilean midfielder Gary Medel for his impact on the game and Mutch for his match-winning strike.
The Scot is content with Cardiff's position early in the season.
"We've got a real belief after six games, about our ability to play at this level and be at home at this level," Mackay added.
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