Horner was speaking after a difficult start to the season for the team in the Australian Grand Prix on Sunday.
Local hero Daniel Ricciardo suffered multiple failures over the weekend. He suffered a gearbox sensor failure before the race, and started the race two laps down after emergency repairs.
A separate issue with the RB13's fuel cell forced Ricciardo to retire shortly before half race distance.
His team mate Max Verstappen fared better and finished the race in fifth place - but he was almost half a minute behind race winner Sebastian Vettel.
"He never gave up, didn’t put a wheel wrong all afternoon and drove a very mature race," Horner said of Verstappen's afternoon. "He did everything that could have been asked of him. He had some brake wear to manage in the last three laps but fifth position today was the optimum result."
The FIA had ordered the team to remove a controversial 'trick' suspension system prior to the race. However Red Bull insisted they wouldn't have used it in Melbourne in any case due to weight reasons.
"In the winter, we had just started to implement and improve it," explained Horner. "Honestly, even if we had the opportunity to use it, it's not a fact that we would have continued down that path because of the weight.
"In fact, the configuration of our suspension now is the same as last season."
Overall, Horner insisted that things weren't as bad for Red Bull as they might look on this initial showing and that improvements would be quick to come.
"We want to progress quickly," he said. "Our car is clearly third, but in the next two races, we intend to reduce the gap.
"In Melbourne Ferrari made the strongest impression and they probably have the strongest car," he admitted.
"But quite honestly, I don't think Mercedes is too far ahead of us. In my opinion we need to add about half a second per lap."