Alright I'm no expert but I'll try to help out as much as I can!
Turtles are a lot of work and it's good to see you are preparing for that. If you are having trouble with what to do exactly and what kind of environment to set up for them, may I recommend a small book that I found to be very useful for beginners? https://www.amazon.ca/gp/product/1882770684/ref=oh_aui_detailpage_o08_s00?ie=UTF8&psc=1 Or if you're not interested, you can ask me and I'll be happy to help however I can!
I dont know your situation and I know it might be tricky to find the time or money to care for them, but they make life long friends and, in my opinion, are very worth caring for. If you end up deciding not to keep them, please don't release them in the wild and try to find a proper home for them.
First of all, 20 gallons is small for two turtles, especially as they grow up. I know a big tank is expensive, but try to look around on sales/trade websites and you might be lucky and find one used and relatively cheap. I got mine, which is a 55 gallons, for around 200$, which is a lot in one go but at least you'll never have to upgrade again. Also, I'm not very familiar with turtle species so I can't help you identify them, but honeslty all semi-aquatic turtles have similar needs. My best advice is to bring them to a vet, if possible, so they can identify them for you and do a quick health check!
As for filters, external filters are the best for turtles, but I also found that having multiple internal filters also works, if this option fits better for you. Once again, used, you can probably find them relatively cheap, but be sure to wash them first! I use a fluvial 106, which is the smallest model of this brand, but if you clean it at least once every two weeks it will work perfectly fine for you. https://www.amazon.ca/Fluval-A202-106-External-Filter/dp/B005QRDDEM/ref=sr_1_1?ie=UTF8&qid=1490500520&sr=8-1&keywords=fluval+106
Making a 100% water change isn't deadly, and sometimes even necessary, but it does damage the habitat built up over time in the tank. I've made a 100% water change ever so often for my turtle and I don't belive he's suffered for it. So, someone correct me if I'm wrong, but I belive you should be fine with doing it.
I've had cloudy water in my tank before, and it is very normal. It's usually white-ish and goes away after a few days. This, however, doesn't look like the same thing, at least from the color of the water in the picture. I think you problems might very well be the gravel you have in you tank. See, this kind of gravel will keep poop and food to rot in between them, and that will cause the water to be very dirty. I'd say take them all out, your turtle doesn't need them anyway, and do a cleaning and water change. I'm 90% sure this will fix your problem. It might get white-ish again but should go away after a day or so. If that is not the problem, then perhaps it's the filter that's dirty and just filtering and pushing out more dirty water. I'd check that too if I were you.
Overall, your current set up is definitely an upgrade from what they had before, but it might need some work. Educate yourself about turtles, it will help! Not only environment, but food, heat, required lighting and potential diseases! And on a last note, thank you for taking care of these little babies. I know it's hard setting everything up and learning about them, but they really are cute pets despite the work. You're giving them a good home and I admire you for that. I did something similar for my turtle, which I took after someone didn't care for it properly and now he has permanent health issues because of it. I still love him and wouldn't trade him for anything in the world though! The world needs more people like you and me who are willing to take care of a small creature in need. :)
Best of luck and, please, feel free to message me if you need anything! As I said, I'm not an expert, but I'd gladly help however I can. Good day!