In an informal style, be afraid is more common than fear.

Don’t be afraid. (NOT Don’t fear.)
Are you afraid of the dark? (NOT Do you fear…?)
I’m not afraid to say what I think. (NOT I don’t fear…)

“I’m afraid” = “I’m sorry”
I’m afraid (that) often means “I’m sorry to tell you (that)”. It is used to introduce apologetic refusals and bad news.
I’m afraid (that) I can’t help you.
I’m afraid (that) there has been a mistake.

I’m afraid so/not are used as short answers.
Can you lend me that? – I’m afraid not.
It’s going to rain. – I’m afraid so.