There is no way to increase volume in-game during the tests, that I know of. If this is a problem for everyone and not just something quirky about your studio setup's current state and/or the sequence of steps you followed, it's worth pursuing.
To figure this out, I'd be interested in which games you played (should still be in your dashboard, for the next part of a day, anyway - just hover over the five icons at the top of the page to see the names.) Also, which instruction video? (use your browser's history, if you're not sure - or use, on SoundGym's Learn page, your place, last eyeball icon in the list - for reference.)
I'm asking for this because I've never experienced what you did (though it's been a while since I've watched instructional videos, after reading your post I did just try the Learn page's Mix & Mastering, "Mixing with Mike's "Fundamentals of Mixing Lesson 16 part 1: Filters and Subtractive EQ" and went to a loud spot in the video and it didn't seem excessive.)
This was just after finishing my workout (and not touching my monitor volume between the SG workout and the instructional video).
My games for the aforementioned workout were, in order from first to last: Peakmaster, KitCut, db King, Filter Expert, and EQ Knight.

At @David Mac, are you using Windows or MacOS. I assume you were using the same browser, even the same tab of the browser for both the workout and the instructional video (you finished your workout in a tab, and then used that same tab to go to the SG Learn page and then played the video?)

AFAIK the levels are set somewhat lower in the games to give room to boosts without harming your ears or speakers. The volume of the videos is managed by your Youtube settings, you can control that.

@Carbon Method - the only Youtube volume control I know about is at the bottom left of the player - which you can only access after you load and start the video. Is there some other Youtube setting where you can control the volume, so it defaults to some lower volume from the beginning of play?
If the Learn videos need to be set, somehow, to a lower volume because they're blastingly loud, even for a second before you can manually slide the YouTube bottom-left volume slider to a lower volume, I'd say that's an issue (though whose fault, I'm not sure.) I'm not hearing that in my studio, and David Mac hasn't convinced me it's not a quirk in his system (though he could if he gives the information I asked in posts above, and we can figure out that's what's up.)

@John Miller I think it is an inherited value, it will pick up the volume set during your last video play. You might need to jump on YT first to have a vid open to adjust the level but looks like working.

@Carbon Method, just tested your "inherited value" idea, which seems to be true (I watched an unrelated YT video, took the volume slider to 50%, closed the tab, then went to the SG Learn page and started a YT video - the slider was now also at 50%.)
Thanks for that, I was unaware.
Still, that doesn't resolve the issue David Mac is having, if it is a legitimate issue. David is not hearing SG games at a level that matches YouTube videos at the default volume slider value of %100. Again, I am not seeing that volume mismatch problem in my setup. (Your comment to David is a useful workaround for him to use - until this problem, if it is one, is fixed - and it should be fixed, if so.)

No worries, mate. I don't read those details from David's comment what you mentioned, let's see what he gets back with :)

Hi all. I've just shot a quick video which you can view here (possible distorted sound warning - it used my display mic for some reason!)
https://www.loom.com/share/fb7d15fe51224bc9ac5fe6cd7a42e784
I realise the volume will likely be lower in-game by necessity as we're dealing with sound monitoring differences, but I really have to crank my volume up to get anywhere near 'standard' volume levels. I'm on a mac, using chrome.

Seems to be normal to me, your YT is cranked up to 100% that gives the bigger contrast in my opinion.
Just keep in mind SG games a bit lower in volume. I use my interface's volume button to adjust differences whatever application I step into use.

I have to move my youtube volume to approx 1/10th of the way up to match levels to in-game volume. I'd estimate the volume difference between youtube at its default 100% and in-game levels to be 3 or 4x quieter. I would expect a slight difference like I say, but I've literally never had any audio app play so quiet in recent memory. I have to turn my interface (apollo) from 9 o'clock, to almost 12 o'clock to match levels. Perhaps the difference isn't coming through as obvious in my video?
Also, John doesn't seem to have the same level discrepancy: "David is not hearing SG games at a level that matches YouTube videos at the default volume slider value of %100. Again, I am not seeing that volume mismatch problem in my setup." I thought that was also interesting.

One more quick thing that may be related, is the volume button on the bottom right a toggle on/off, is it meant to be a slider of some kind? I only have the ability to toggle it on or off, which i suspect is the correct behaviour. The reason I mention it is when I do click it, I seem to get some kind of graphical glitch when I click it a second time to switch volume back on - it turns into a barely clickable horizontal line. (see image)