McLaren went three rounds in 2023 before scoring a point as it struggled to find competitive form out of its car.
But a raft of upgrades brought for both Lando Norris and Oscar Piastri in recent races has allowed the former to score consecutive second-place finishes, while Piastri was in the top five.
Albeit some 136 points adrift of Mercedes in second in the constructors' standings, Russell believes McLaren is a genuine threat in the best-of-the-rest fight behind runaway leaders Red Bull.
"It's absolutely clear they are true contenders for the second fastest team," Russell said of the Woking outfit.
"It's very odd how Aston Martin were clearly that second-best at the start of the year.
"And they don't seem to be so competitive right now. Ferrari haven't made much progress. And McLaren have made a huge jump.
"So if it wasn't for McLaren, we'd be very, very satisfied with the progress we're making. Moving ahead of the midfield pulling a gap on our rivals, closing on Red Bull.
"McLaren have just totally thrown it in the mix. But it gives you optimism that there is potential to make bigger steps.
"I believe in my team. I think it gives us confidence and optimism that we can make that step to Red Bull.
"We're not too focused really on McLaren or Aston or Ferrari. We're focused on Red Bull. And we're trying to make that big step."
Russell enjoyed a strong fightback from 18th on the grid in last weekend's Hungarian Grand Prix, vaulting up to sixth at the chequered flag having been told by Mercedes' strategists that 11th was looking most probable.
While pleased with his end result, Russell believes it was "proof" that Hungary was a "missed opportunity".
"The strategies were telling me this morning P11 is most realistic and P7 if we maximise everything," said Russell when asked by Autosport/Motorsport.com.
"To come away P6 with no safety car no VSC on merit was a really great result.
"But equally proof that this weekend was probably a missed opportunity. I'm confident I could have been up there with Lewis yesterday, this is one of my favourite circuits, the car always performs well here.
"And when you've got two cars up there, fighting for P2 gives you a lot more options, and Lewis was really strong as well. And if things panned out slightly differently he would have been P2 as well.
"So as a missed opportunity, we'll learn from it. But the positives are we're moving ahead of Aston and Ferrari."
