The word "fight" means to battle or stuggle against something. The word "fighting" in English is not the same as 加油, because "fighting" can not be used on it's own without a helping verb. "fighting" is the present participle of the word "fight." "fight" can be used, in some contexts, to mean ”to do ones best," but it is not used by itself as a cheer. You can only describe a situation:
For example: He is fighting to become the best athlete of his generation.
If you want to cheer for someone, you can say something like "do your best!" ; "You can do it!" ; "Come on!" ; etc.
I hope this helps you. If you have further questions, please let me know.