It is ok, but it would be incorrect if you used what Karl suggests. I think that you don't mean "to study" Turkish as your profession. So, you can't say "okuyorum" instead of "çalışıyorum". "Çalışıyorum" is fine in your sentence.
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you can also say; Türkçeyi kendi başıma öğreniyorum.
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yes that's right. you could also use okuyorum instead of cailsiyorum.