Though both parties have made an underwhelming start to the 2015 FIA Formula One World Championship, Horner says the improvements seen at the last round Bahrain - and the gains made over the extended break between races - could pay dividends in Barcelona.
"It's been a really good opportunity to re-group after the early flyaway races, look at all the information and data gathered from them and of course get ready for the start of the European season next week," he told the team's website.
"[The first four races were] far from where we expected to be and ultimately disappointing, but what we can see is we're making progress - I think Bahrain was probably our most competitive race so far.
"We can see on the engine side that Renault are making some steps forward as well. I think in this two week period to re-group, and Renault to have done plenty of running on their dynos and iron out some of the issues they've had, has been very timely. Hopefully we can use that to great effect in Barcelona next week."
Horner also said Red Bull can draw inspiration from their remarkable turnaround in 2014, when they put a disastrous pre-season behind them to emerge as Mercedes' main threat over the opening half of the season.
"After a difficult start last year we bounced back from pre-season testing to the first race and we want to try and make a big step forward in the next few races to put Williams and maybe even Ferrari under a bit of pressure," he added.
"The team has been used to winning over the last few years and it's tough when you're not running at the front where that's where you're used to being. But the great thing here is that everyone is using it as a motivation to get back there.
"Everyone has got their head down, everybody's working incredibly hard and we can see progress in many areas."
While Red Bull are currently fourth in the constructors' championship, they have collected just 23 points - their lowest haul from the opening four races since 2008, when a different scoring system was in use.