Is it normal for a modified Supra to overheat after 10 laps?

Ran my tuned Supra at track days for about three years and what you're describing is textbook heat soak, not necessarily a fault. These B58s make easy power with a tune but the cooling system is sized for street duty, not ten flat-out laps. Once the stock radiator saturates it can't shed heat fast enough and temps climb quick, especially on tracks with long straights where you're buried in throttle for extended stretches. That said, have him check the cheap stuff before spending money. Pop the front end apart and look for missing or misaligned ducting, a cracked shroud, or anything blocking airflow to the core. Also bleed the system properly if anyone opened it up during the build. Air pockets cause exactly this behavior. And check whether the tune raised the cooling fan threshold, some tuners bump it higher than you'd want for track use.

If all that's fine, then yeah, the upgrade path is what you'd guess. An upgraded radiator helps a lot and is usually the first thing to do. But honestly on a track-only-ish car I'd do the radiator and an oil cooler together. The oil temps were my actual limiting factor, water temps came down with a bigger core but oil kept creeping past 280F til I added the cooler. High oil temp means the ECU starts pulling timing and the car feels sluggish by lap 8 even when water looks okay.

Driving style matters too but it's a band-aid. Short shifting and coasting down the straights will keep temps in check but you're leaving lap time on the table. Better to fix the cooling and drive the car like it's meant to be driven. One habit worth building regardless: do a slow cool-down lap, no braking hard, then let it idle a few minutes in the pits with the heater blasting before shutting off. Heat soak right after shutdown is what warps stuff and cooks gaskets. Those SuperRace cars are interesting but they're running short sprint formats, a track day with 20 minute sessions is way harder on the cooling system.

One thing I'd add to the oil temp discussion is the diff. Ran a tuned coupe at a couple of track weekends and the rear end got scarily hot, no cooler on it at all. Never hit a limp mode or anything from it, but it made me nervous enough to let the car sit longer between sessions. Worth checking the diff temps if he's planning longer runs, especially if the car still has a stock setup back there. Curious what the SuperRace cars acutally run, they must have something engineered for it given the sprint format they use.

The stock radiator is definately the first thing I'd swap, but don't sleep on the airflow side of it. I run a shop that does track prep on BMW platforms and I've had cars come in with overheating complaints that were fixed with nothing more than repositioning the bumper ducts and sealing gaps around the core support. Air takes the path of least resistance, so if it can leak around the radiator instead of going through it, you lose a shocking amount of cooling at speed. I had one M340i in here last month that was heat soaking every session, and just fabricating a proper air dam under the front brought the water temps down about 15 degrees without touching the radiator at all. Tell him to eyeball where the air's actually going before he drops money on hardware.

Agree with the cool-down lap advice, that habit saved me more than once. I also got in the routine of logging temps on every session with a cheap OBD reader, so I coud see exactly where the climb started instead of guessing. Turns out mine always spiked in the last two laps after back-to-back runs, which made it obvious the system just needed more recovery time between stints. Helped me plan my day around it too.

I don't have a Supra, I track a turbo Miata, so take this with a grain of salt, but the session length thing hits home. Mine never overheats in short stints either, it was only when I started doing full 20 minute runs that the temps got away from me. One thing that helped me was adding a second fan switch point so the fans actually run on the cool down lap, not just at idle in the pits. Not sure if the BMW setup lets you do that easily, but it made a difference for mine. Also curious, did your buddy's car run hotter in the summer heat? Mine is way more sensitive on hot days and I wonder how much of the problem was just the weather that day.