When somebody says: "Do you want me to help you with that bag?" Can I answer: "No thanks, I'll manage or I'll handle it"
Yes, you may use either 'manage' or 'handle' here.
I agree with abdmlkbd that either way is fine.
[intransitive] to succeed in dealing with problems, living in a difficult situation etc
- I don't know how she manages with seven children.
- We didn't have the proper equipment, but we managed somehow.
"Do you want me to help you?"
"Ah, no thanks, I'll manage."
[transitive] to deal with a situation or problem by behaving in a particular way and making particular decisions
- Leave it to me. I can handle it.
To the enquiry
"Do you want me to help you with that bag?"
A native speaker would reply:
"Thanks, I can manage it."
"It's OK, thanks, I've got it."
Yahoo!?? - ????
"Do you want me to help you?" "No, thanks, I can manage (it)."