I currently have 4 small comet goldfish in a 10 gallon tank...I don't know what to do, I've only had them for about two weeks.
Well, the first thing you should do is become accustomed to seeing and identifying fecal matter in your goldfish tank.
I'm doing 25-~50% water changes weekly, as my filter isn't strong enough to keep up with their waste output...
Kidding aside, your brother's betta has shown you the likely outcome of not changing water in a tank that is apparently going through a fish-in cycling process.
Your comment above seems to indicate that you think water changes are optional, and that you are doing them only because your filter "isn't strong enough". You need to get over that idea right now; any filter you are likely to get for your 10-gallon tank is not going to lessen the need for water changes. You need to do some research on and understand the nitrogen cycle, and to begin cycling your aquarium...which means that if you expect those fish to live in that tank, you can probably expect to do 50% or more water changes several times weekly, possibly even daily, until your bacterial colony establishes itself and begins to consume ammonia...probably several weeks at least. The only way you can monitor the progress of this is to have a water test kit, and use it carefully and according to directions.
You need to feed as little as possible during this period, in order to minimize the chances of an ammonia spike. And, honestly, that coarse gravel shown in both of those tanks is the worst choice imaginable. The individual pieces are too large to allow the goldfish to indulge their natural instinct to sift through the substrate for food; but having a thick layer like that simply creates endless nooks and crannies into which uneaten food will settle and decay, beyond the reach of the fish. A thin layer of sand, or a bare bottom, is much easier to keep clean, and won't conceal waste.
While that unfortunate betta is toast...your goldfish really don't look bad and with proper care should be able to survive the coming weeks, but make no mistake: a fish-in cycle is a stressful and dangerous process for any fish, and this is especially true when talking about 4 (!) goldfish in a 10-gallon tank.
Good luck...you are going to need it, and so will those fish...but be diligent and perhaps you can save them. And...start planning on either re-homing them or getting a bigger tank. This isn't "eventually" or "next year" or "down the road" either. Don't wait until you think they have outgrown that tank...because it's too small now, and they will grow fast...if they live.