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I bought a cheap Bio-Foam Filter. It says that it is best for small tropical fish tanks. I have a 15gallon tank. I don't have anything yet in the tank. Just the turtle and the basking area yet the filter can't even clean it. The water is not clear and the dirt from the turtle are still in the tank floor. I don't even know if this filter sucks any dirt at all. Suggestions? I don't have that much of a budget to buy the best filter.

Your turtle is going to grow fast and you will end up at some point with a large tank and probably a large canister filter. Anything you buy now will be an interim solution. You either want to buy big now OR minimize your spend on something temporary until you get the large setup.

For something less expensive for a juvenile turtle:
You might check out one of the aquaclear or similar hang on type filters. You might also look at one of the generic canister filters off ebay like the sunsun. Either can be had for around $40 US dollars. The sunsun canister should get you a couple of years and up through a 40 gallon breeder sized tank.

Okay canister filter. I also saw that undergravel filters can be used but most of the people commented that they are not safe for turtles. Is this true?

Well, gravel itself is not safe for turtles...

I now have an aquaclear hang on type filter. Would this suffice for the RES turtle?

A very large aquaclear will work with a turtle up to about a 20-30 gallon tank yes.

I also bought a aquarium liquid filter. It sucks dirt fast from the tank's floor or even in the water then brings back clean water after filtering.