McLaren started the season off the pace but following a change of concept and personnel reshuffle it has produced an impressive B-spec that turned the MCL60 into a reliable podium contender.
Norris and team-mate Oscar Piastri became the closest challengers of Red Bull's dominant Max Verstappen in Japan and Qatar, taking a double podium for the Woking team while Piastri also snatched a sprint race win in Losail.
That allowed McLaren to get within 11 points of fourth-placed Aston Martin in the standings, while even the third place of Ferrari appears to have come under threat, with the Scuderia now 79 points clear.
When asked if Ferrari can still be caught across the remaining five weekends, which include two more sprints, Norris replied: "It’s doable, I think so. A couple of races ago it was 70-something [points] to Aston.
"I think we're just doing a good job, not just in terms of pace but we had a 1.8 second pit stop which is pretty impressive from the guys. They've been working extremely hard so to reward the whole team consistently is only more motivating and a bigger boost every weekend."
But Norris also cautioned that not every upcoming circuit will suit the McLaren as much as its preferred high-downforce swoops of Suzuka and especially Losail. Therefore it might not be able to keep taking home big hauls such as the 33 points in Japan and 47 units in Qatar, which included a sprint.
"We know that Mercedes are very quick, probably almost as quick as us [in Qatar], just they keep making mistakes," he explained.
"So, I'm sure if they have a clean weekend they will make our life a bit more tricky, and therefore I'm not as confident in saying it.
"It was only three weekends ago that Ferrari were on pole and winning races, and nothing has really changed, it's just a different track. We're looking better, they’re looking a bit worse.
"I think we're definitely not overconfident. We're confident, I think we need to be, but we know there are still going to be some races where we're not going to be quite as strong."
Piastri, who took both his first F1 podium and sprint win in Suzuka during an impressive rookie campaign, felt catching Ferrari will be a "big challenge" but not an impossible one.
"Suzuka and [Qatar] have probably been tracks we've been looking at for quite a while in terms of what's going to favour us," the Australian added.
"There are going to be some races that are going to be more difficult and probably will suit other teams a bit better.
"I think 79 points in what, five races and a couple of sprints? It's a big challenge but we'll give it a go. Hopefully we can do it."
