Why is it "j’ai peur" instead of "je suis peur"?
And also is it okay to say "avoir peur"?
First of all, avoir peur in French mean being afraid in English.
The definition of "afraid" is the following:
feeling fear or anxiety; frightened.
So the most important is to understand this is a feeling, which mean we won't use to be / être to express it.
In French (in general cases) we use:
Verbe être / to be when talking about a condition.
Il est malade. He is sick.
Ils sont fatigués. They are tired.
Verbe avoir / to have when talking about a feeling.
Ils ont faim. They are hungry.
Elle a peur. She is afraid.
So in conclusion we use "j’ai peur" instead of "je suis peur" because "being afraid" in French is considered a feeling and not a condition.
What you are trying to translate is "I am scared." or "I am afraid.". In French you do not use "to be", that is "être", instead it's always "avoir" ; therefore you say this;
"Je suis peur." is always incorrect.
The difference is that in English you use an adjective (afraid, scared), which is an estate, that of being overwhelmed with fear in this case, whereas in French you use the corresponding noun instead (la peur). Obviously, you can't say "Je suis la peur." or "Je suis peur.". So, what you say is something as "I have in me the feeling of fear.", somewhat shortened (I have fear.).