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 one minute 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.
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.
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.
Russell enjoyed a trouble-free first session as he lapped with a best effort of 1:16.363, 0.203 quicker than Piastri.
First practice saw a cast of reserves on track with McLaren's Leonardo Fornaroli fifth in Norris's car.
Others in action included Paul Aron in an impressive sixth for Audi, Dino Beganovic at Ferrari, Ayumu Iwasa at Red Bull and American Colton Herta, a nine-time winner in IndyCar, at Cadillac.
Luke Browning had been due to take Alex Albon's Williams for the session, but failed to set a lap time due to an electrical issue.