Frenchman Gasly, who replaced Daniel Ricciardo at the team, crashed twice during pre-season testing in Barcelona last month, restricting team mate Max Verstappen’s time on track.
Ahead of the start of the new season in Australia next weekend, Gasly is already under pressure to prove he can compete for his new team.
"The team is very busy, with the first race being on the other side of the world, 10,000 miles away, in Australia,” Horner said at an event with new engine partners Honda in Tokyo.
“Pierre has kept the team even busier after his incident last week.”
Gasly, who was seated next to Horner, said: "I need to learn from these mistakes."
"It wasn’t the best end of the two weeks but I will remember the positives and work on the negatives.
"I think overall, the package, the first feeling with it was great.
"Obviously, I am not feeling as comfortable as I would like with the car and it is my job to set it how I want with the team around me."
Verstappen and Gasly also thrilled fans by performing doughnuts on the streets of the Japanese capital during a Red Bull promotional event. The drivers were in Tokyo in support of Honda, who will be supply the team with power units this year.
Red Bull parted company with engine partners Renault at the end of 2018.
Horner said the new relationship with Honda was blossoming and that the team were ready to challenge Mercedes and Ferrari in the championship.
“It is the start of a new era for Red Bull Racing with Honda,” he said.
“Everything we have seen from the power unit so far, it’s reliability, it’s performance, the way it is incorporated into the chassis, is extremely encouraging.
“We have had a positive pre-season and are hopeful and optimistic for the season ahead.”
Red Bull, who last won the championship in 2013, finished third overall last season, over 200 points behind winners Mercedes.
“Mercedes, having won five consecutive world championships now, and Ferrari are currently the benchmark,” said Horner.
“But we are confident we are closing that gap and we have all the tools around us to be really able to take the fight to both of those teams and manufacturers.”
The team will now travel to Melbourne ahead of the Australian Grand Prix on March 17.
(Editing by Peter Rutherford)
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