The 19-year-old lost control of his car approaching Turn 13 and slid sideways into the barriers. The impact damaged both the front and rear suspension, while his front wing was also broken.
Despite the crash, Lindblad finished the session fourth on the timesheets, just 0.228 seconds behind Mercedes driver Kimi Antonelli, who set the fastest time around the new Madrid circuit.
Ferrari's Charles Leclerc was second fastest, with team-mate Lewis Hamilton third. Both drivers were just over a tenth of a second behind Antonelli.
Mercedes' George Russell finished fifth, ahead of Red Bull's Max Verstappen and McLaren's Oscar Piastri. Russell currently trails team-mate Antonelli by 66 points in the championship standings.
World champion Lando Norris was unable to record a representative lap after a gearbox problem kept his McLaren from setting a time during the session.
Lindblad's crash was the only major incident during Friday's Formula 1 running, although it highlighted concerns raised by Russell before the session about the demands of the new Madring circuit.
Russell had warned that the track offered "less margin than Monaco" because of the higher speeds involved.
He explained that while Monaco's barriers are closer, drivers travel much more slowly, giving them a better chance of recovering from small mistakes such as a lock-up.
Russell identified Turns Five, Seven and 17 as particularly challenging sections. Leclerc ran wide at Turn 17 and briefly entered the run-off area during the session.
The Mercedes driver also predicted that the circuit could produce plenty of lock-ups, particularly at corners where drivers have to brake while turning. He compared one of these sections to Turn 15 at the Baku street circuit, where drivers have previously struggled to slow their cars before reaching the barrier.
Russell also pointed to a series of high-speed direction changes that reminded him of the fast sections of the Jeddah street circuit in Saudi Arabia.
With the new Madrid venue proving demanding from its first official running, drivers will be looking to balance outright pace with caution as the Spanish Grand Prix weekend continues.
