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"I've received good support from the owner, chairman [Mehmet Dalman] and chief executive [Ken Choo]," he said.
"They know what we're trying to build and create.
"Of course there's a need for short-term results, we don't want to be in the league position we're in because it hurts.
"Patience is key, especially in the modern game, because everyone wants short-term results, I certainly do.
"It hurts me that we're not getting them."
The defeat by Leeds was Trollope's eighth league game in charge of Cardiff, having succeeded Russell Slade at the end of last 2015-16.
Even at this early stage of the campaign, however, the future of the former Bristol Rovers boss has been the subject of scrutiny.
"The results are nowhere near acceptable, that's my responsibility, I know that," said Trollope.
"It's my responsibility to shape it and mould it and get the best out of people.
"I'm totally aware of that, but I also believe in how we work, that this will turn."
Leeds secured victory with an assured second-half display, though their opening goal came against the run of play after Cardiff had controlled the game before the interval.
The visitors were awarded a penalty when Bluebirds defender Matt Connolly tangled with Pontus Jansson at a corner and wrestled him to the ground.
Leeds manager Garry Monk said it was a "definite penalty" but, having seen players grapple each other from set-pieces throughout the game, Trollope felt the decision from referee Graham Scott was inconsistent.
"Probably the refereeing decision changed the game, which was frustrating for us because there was a lot going on at set-pieces at both ends," he said.
"There was a warning before. Listen, I'm not saying it wasn't a penalty, but if he's giving it then he could be giving a few at either end.
"That moment changed the game and meant we had to chase it in a different manner, we left ourselves a wee-bit open."
Things are looking much better for Garry Monk. The former Swansea manager has had back to back wins this week to ease the pressure on his position.
"It's been a good week - today was another step forward. It was a good team performance." Monk told Sky Sports.
"We deserved the win today. We were clinical when we needed to be and defended well when we needed to.
"I'm pleased for the team. Getting back-to-back wins in this league is vital.
"Our determination in the second half got us the win. It was a penalty but a goal was coming.
"There's nothing better than giving these fans an awayday like that. Everyone played their part today.
“All I could hear was our fans. Right from the start they took control of the stadium, it was never about me and my history.