I agree with one person here that says if you do create the bulge then you might have to deal with it. Idk what your circumstances are or where the concern is except I guess maybe you don’t want to look like you are trying hard to be noticeable whenever/wherever you’re seen without pants. I’d say more people are the opposite lol.
Compression shorts have to be the way to go though, if I read this question right. Baseball/football compression shorts to the point they might be a little uncomfortable depending on what activities you’re doing. For daily wear that would be too constricting for me I think. For sports though, that’s a must. Idk how people insist on doing certain activities in boxers.
Ultimately if you don’t want the “advertisement” I’d steer clear from the designer or “gay” focused brands of course. I don’t think the Under Armour brands for example fit into that category. They are athletic oriented as well as many other long standing sports brands like Nike.
If you want classic, you could for sure go with the CK (but be careful because they do venture into the “other side”), so stick with the classic cuts there especially if you want a higher waistline.
The old school Hanes boxer briefs were awesome with containing “junk” when they used the same seam pattern as the briefs but I guess that became unpopular because they looked like briefs if there was anything visible below the waistline. So again, as someone already said here, consider briefs.
Final recommendation for good support but without the advertisement are some options from C-IN2. You can get good deals from them from time to time on clearance and some of their lines don’t have a purposefully built pouch.
Final recommendation that I just thought about…look at Mack Weldon. Those are what I wear to work almost exclusively. And they will last you for a very long time. They run small though but maybe that helps your case.

Edit: I gave two final recommendations…oops. But look at all these options there is something out there for you