McLaren emerged from Bahrain's season opener as the fourth-fastest team, well behind Red Bull and Ferrari on race pace while nipping on the heels of Mercedes.
And while its form compared to Mercedes is still too early to call definitively, with Bahrain a tough circuit for McLaren while Mercedes suffered overheating issues, Norris believes the team will need two big steps to catch up to Ferrari and then the dominant Red Bull.
"There's a big step between us and Ferrari on a circuit like this and then another step from Ferrari to Red Bull," Norris said after finishing sixth at Sakhir.
"We've got two big steps to make up but starting with one would be good."
When asked if making those steps feels achievable this year, he replied: "To Ferrari? Yes. Again, at certain tracks, we're going to do it and at certain tracks, we're not."
In the meantime Norris expects Ferrari to put some pressure on Red Bull on occasion despite Max Verstappen's Bahrain dominance, pointing out Charles Leclerc's Q2 lap would have been good enough for pole.
Asked if Red Bull was on another planet, he replied: "I mean, they weren't. Carlos [Sainz] was not far behind.
"I know the Red Bull is very good on race pace and on this track, you need to be good on tyres, because it's so abrasive.
"They clearly are still ahead, that's for sure. I think when you look at qualifying, they weren't obviously ahead.
"I think Charles had the quickest lap, so maybe between them they can battle a little bit."
Team principal Andrea Stella felt Ferrari and Mercedes were "within reach", but preferred to wait until this weekend's Saudi Arabian Grand Prix at the vastly different Jeddah Corniche circuit to draw more definitive conclusions about the 2024 pecking order.
"The gap to Max, it's not necessarily the main parameter we look at. I'm looking more at the gap to Mercedes and Ferrari, which seem to be within reach," he said.
"Let's see where we will be in Jeddah. That should be a more favourable circuit lay-out for our car and between Bahrain and Jeddah we will have a proper assessment in terms of our competitiveness.
"We have the Mercedes, then the Ferrari and [Sergio] Perez, then Max. There's some steps before we think about the victory, but those steps are not too big, which is an encouraging aspect that we take away from this weekend."
Key to its fortunes will be better low-speed performance, even if the team has already improved "at every speed."
"We did an overlay of qualifying in 2023 and in 2024 and we go 1.8 seconds faster," Stella said. "We improved at every speed.
"Even in low speed, we are faster because we added downforce overall, but most of the downforce was added in the medium range.
"The low speed still deserves some specific work and this is what in particular our aerodynamic group is working on."
