Norris saw both of his Q3 laptimes deleted in Friday's qualifying at Losail, meaning he would start the race from 10th rather than second behind polesitter Max Verstappen.
In Saturday's sprint shootout, Norris then went off track in the final corner of his second flyer while on a pole-worthy run, instead getting demoted by team-mate Oscar Piastri.
Piastri duly won Saturday night's sprint race from pole, while Norris recovered from 10th to third in Sunday's grand prix thanks to McLaren's rapid pace on its favoured high-downforce Losail circuit.
Norris was hard on himself after his qualifying mistakes, saying "I am so sh*t sometimes" on the team radio on Friday because he felt he should have been able to fight for not one but two victories in Qatar.
"I don't know how much quicker Max could have gone in qualifying, but I guess if I want to be honest with myself, I should have fought for two pole positions this weekend and potentially two victories," he said.
"I know it's a big thing to say, but if I just did a better job and didn't make the mistakes ... which is why I've been so frustrated. Because I knew what was on the cards, what we were able to potentially achieve and that was [two] victories.
"So, I feel like it was a weekend of missed opportunities for myself to take the fight to Max and at least give ourselves that opportunity.
"I'm definitely still happy with a P3, but in the back of my head, I know I should have done more this weekend."
Norris finished in the wake of Piastri as McLaren tied up a second consecutive double podium behind Verstappen.
The team finished where it was on merit, although it was aided by Mercedes' Lewis Hamilton clashing with his team-mate George Russell at the start, which ended Hamilton's race and also knocked Russell out of podium contention.
Nevertheless, Norris' comeback from starting in 10th to finishing just five seconds behind Verstappen was what gave him the confidence that McLaren's "better than expected" race pace could have yielded a better result with a front-row start.
"We knew we had great pace, maybe not initially to fight for a win, but to get up to the front at least and to fight the Mercedes, but they did a lovely job of making our life a lot easier," Norris added.
"By the time we came through Turn 1, I felt like we could at least fight for the podium straightaway. So from lap one a podium was on the cards and that was my target.
"But the pace turned out to be probably better than I was expecting, for us to be at points in the race probably a bit quicker than Max, maybe at some points a bit slower.
"On the whole, to finish P3 was as good as it could have been [from 10th on the grid]."
