I agree that there isn't one single way to say this in English.
"Come on" has so many meanings that can change with the tone of your voice. If you say it wrong it could sound strange, or even insulting. Also this use of "come on" is very uncommon in American English.
At a sports event, you can say, "Let's go [team]!" or "Go [team]!"
If you want to say 加油 to a person who is having trouble, you can say, "Don't give up!" or "Keep trying!"
For a person who is doing well, you can say, "Keep up the good work!"