Russell was fastest in a hot and sunny first practice but Norris, who sat out that session, was 0.009 quicker in the second with the day's best lap of 01:15.426 seconds.
McLaren's Oscar Piastri, last year's winner from pole position in a McLaren one-two at the Circuit de Catalunya, was third and Ferrari's Charles Leclerc fourth.
"Today was a reasonable Friday. Not perfect, but a step in the right direction," said Norris. "The heat and wind made it tough for everyone, so no one's feeling completely comfortable out there, but our baseline is stronger than it has been at previous events.
"We've come to a circuit that certainly suits us better and the car is operating in a very different window, which is encouraging."
Mercedes's championship leader Kimi Antonelli, who also sat out the first session while the team gave track time to reserve driver Frederik Vesti, was fifth as the 19-year-old Italian prepared for a weekend that could bring him a sixth win in a row.
"Overall, still quite a bit of work to do," said Antonelli. "But I'm quite confident ahead of tomorrow. The long run was good, so that's a positive. I'm looking forward to the changes we are going to make."
Red Bull's four-times world champion Max Verstappen was sixth fastest, with Racing Bulls' Arvid Lindblad seventh and Audi's Gabriel Bortoleto eighth.
Ferrari's Lewis Hamilton, Antonelli's closest championship rival but still 66 points behind after finishing runner-up in the last two rounds, was ninth.
"We had an issue with the front wing which made things a bit more complicated, particularly in terms of tyre preparation, and missing FP1 (first practice) meant I was a bit on the back foot," said Hamilton.
Russell enjoyed a trouble-free first session as he lapped with a best effort of 1:16.363, 0.203 quicker than Piastri.