There isn't any turtle that can stay in a 20 gallon it's whole life, but there are some species that are "smaller" in comparison to other species.
Map turtles (there are many species of map turtle, for example, the mississippi map turtle) are smaller, the males will grow to be about 5 inches in diameter, while the females will grow to be about 12 inches. The rule of thumb is that a turtle needs at least 10 gallons of swimming water per every inch of shell.. for example, a 5 inch turtle needs a 50 gallon tank, and a 12 inch turtle needs a 120 gallon tank. You can buy a Mississippi map turtle for about $25 on the internet, but you won't know if you're getting a male or female.
I believe spotted turtles also stay small-ish, adults of both sexes growing to about 5 inches, and they are very unique, although more expensive to buy.
There is a lot of turtle-specific info to learn, but it just takes a little reading and practical application. If you decide you want a turtle, PM me and I can answer whatever questions you have, or direct you to a fine turtle forum.