McLaren has taken a major step with a recent string of upgrades for the MCL60, and in both Britain and Hungary Norris led the chase of Red Bull and earned back-to-back second places.
He acknowledges that the team can now expect to challenge for the top three on pure performance rather than having to wait for unusual circumstances to create a chance.
"I think the majority of the time we've had an opportunity to be somewhere or achieve something we've been able to do it," he said. "And therefore I'm happy with it.
"It's just adjusting to the new reality of it, which I think we've done well as a team. It's easy to get a bit overexcited now we're fighting for podiums all of a sudden when we haven't for a long time, and we definitely haven't consecutively.
"It's more like halfway through the race we're like, 'Oh my God, we're going to get on the podium here,' rather than starting P2 and aiming for a podium.
"It is quite a different story that you're in. But it was treated very well. The mechanics, the engineers, everyone. So I think everyone adjusted well to that."
Norris expects McLaren to bounce back at this weekend's Dutch Grand Prix after the "outlier" difficult Belgian GP ahead of the summer break.
His team-mate Oscar Piastri finished second in the rain-affected sprint in Belgium. However, in the main race, Norris could not better seventh as McLaren lost out to cars from the Ferrari, Mercedes and Aston Martin teams.
"I think obviously Spa was not a bad one for us still," he said. "We just didn't have a race car, we had more of a qualifying car. But even the race car was not bad when we were in clean air.
"It just wasn't a raceable car, I think is the way of saying it. I got overtaken every single lap because we were so slow on the straights.
"But when I was in clean air on the soft tyre the pace was good enough. Obviously, Oscar was one hundredth off [the sprint] pole position. We still fought for good positions, and had a good qualifying.
"This track is a little bit more back in line with Budapest, so I would say there's no reason why we can't continue where we were prior to Spa.
"And I think Spa was a bit of an outlier with efficiency and things like that, which we just didn't have.
"Because so much time and effort has been put into making the car good for 90% of the circuits rather than the other 10%, which includes Spa and kind of maybe a bit of a Monza scenario, which I think is a perfectly valid and excusable thing."
Norris stressed that the fight is so close behind Red Bull that tiny margins can make a big difference.
"I know my weekend in Spa was not to the level it should have been," he said. "I damaged the car in the first quali, and just made one mistake on my second lap. And that made me look a lot worse than what it was.
"The thing is everyone's very, very close in qualifying. Even last place is not that far away. In Budapest, four-tenths or two-tenths or something covered a huge amount of people.
"Everyone closes in qualifying, with new tyres and low fuel making everyone look good. It's then what you can do into the race, which is a much bigger question. That's where Red Bull are just in a different league."