What's the mainboard's exact name/model + bios settings?
I assume that your card is a VesaLocal bus version...
Having a soundcard "running" can take some steam out of a 486.

Sorry for the late response. The motherboard is a UNI486-3SLOT B43003-A. The bios settings were pretty limited but these settings gave me the best performance:
Cache: Internal & External
Shadow: System Bios
256k Reclocation: No
There were also settings to shadow memory at certain addresses but that didn't seem to help. The S3 is just a chipset integrated into the motherboard since I have no VLB slots. I don't think the soundcard was the reason for the performance issues since I noticed that I got a lower score in speedsys. Would it be possible that I'd need faster ram or cache chips? I noticed that my 16mb stick is 70ns and my cache is 20ns.

Better post a screenshot of speedsys and maybe a shot of the board.
Everything that follows is based on vague ideas since I do not know the board's specs...
-the board's S3 chipset may internally be connected via a slow ISA bus(better knowledgeable board members may have more info here)
-maybe the board's cpu multiplier needs to be properly configured?
-does the board contain a slot for a riser card?
-the faster the ram/cache the better, but it's doubtful that inserting higher quality components will drastically increase your score since the core problem seems to lie elsewhere
-the minimalist options in the BIOS aren't helping either. On my GA486-VS8, for example. the 2nd level cache has to be set to writethrough or its performance sucks(35 vs 93 MB per sec). I do not know however if I'm missing something or the if it's the board's fault

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Do you have a turbo switch ? Maybe this is causing some problems ?

I checked over all the jumpers and everything seemed to be setup properly. It has a riser with 3 ISA slots and another slot on the back, but I'm not sure what it is. It looks identical to the slot the riser connects into. There doesn't seem to be any turbo button either. If the S3 has a slow bandwidth would there be any good ISA video cards that would give a better result? Also, here's the pictures of my speedsys score and the board.

It seems the CPU runs as fast as it can. But the motherboard/chipset is slowing it down. The main memory throughput is low and VGA card is connected to ISA bus. You can't expect better results.

Damn, that's a bit of a bummer. But still, thank you everybody for the suggestions!

If your bios has option for RAM timings, try to lower them, that might help

As already said - the bus is ISA and simply "destroys" any performance the CPU could provide. The Riser slot gives away that it's 16 bit -> see attachment for a 32 bit "riser" slot on a motherboard of similar format(taken from an "ad" for an IBM board).
Ram throughput is exruciatingly slow but maybe you can fiddle around with JK5 which goes from 25-66mhz... although I doubt that this is really the board's bus speed.
Your ISA card delievers approx. 3+ MB throughput which is pretty good - theoretically a 320x200 screen could be refreshed over 45 times a second. Have you tried running any game yet to gauge real world performance/usability... other than the Doom bench?

The bios doesn't seem to have any options for timings, so it's extremely limited compared to a standard 486. I took a quick look at the config map for the board and it looks like JK5 is already set up as 66mhz. Other than Doom I tested Quake which ran at 8.7 fps using the timedemo command. One odd thing I noticed is despite the fact the 586 gave better results in benchmarks its memory throughout and bandwidth is slightly lower than the standard DX2's, though I have no idea why that would be.