In practice people normally just say I'm 20.
If you specifically wanted to mention 21 (because it sounds like a more "important" number, perhaps), the idiomatic standard is I'm in my 21st year (which you can reasonably say at any point between your 20th and 21st birthday), but it's a bit pretentious/poetic for most contexts unless "facetious".
If you specifically wanted to convey that you're nearer to 21 than 20 (i.e. - your 21st birthday is less than 6 months away), you can say I'm 20 going on 21.
To convey that your 21st birthday is imminent, you can say I'm almost / nearly / just under 21.
You can say I'll turn 21 next month or I'll be turning 21 next month, but you'd rarely be in a context where you could use the "true" continuous. For example, How old are you? I'm turning 21 would be a rather strange thing to say. The only context where that would seem natural to me is if the speaker doesn't know exactly when the relevant birthday occurs I haven't seen your son for years - surely he must be turning 21 by now.