well im on the swim team so now i wear speedos. They aren't thin side speedos though but theres def tight and cause a bulge.
if im going to the beach ill wear regular swim-shorts most of the time with a pair of boxer briefs underneath them.
You should take a pic in your speedos and post it for us to see!
Why don't you just wear a speedo under your swim shorts? It would be more supportive and would dry quicker.
And if your wearing regular swim shorts with a liner you really don't have to wear anything under them at all, eventhough I always wear a speedo underneath just in case I get motivated to take the trunks off sometime during my day at that pool, waterpark, beach, etc.
Note: If doing this I recommend a conservative speedo with 2-3 inch sides in either a blue or black color, or I have some from the days when they did cool patterns on the entire suit that do a good job to make the package less visible making them appropriate for any situation.
Random Additional Question:
How big are you? Are you Cut or uncut?