My grandma would prevent me from bathing in the beach when I was a child because she was really afraid that I **would** drown. we need would + [present simple] because drowning is an unlikely event.

I agree with Mike that you definitely don't 'bathe in the beach', and even 'bathing in the sea' is little old-fashioned-sounding. A more natural way of saying this would be 'My grandma wouldn't let me go in the sea when I was a child because she was afraid that I'd drown.'

"My grandma would prevent me from bathing at the beach when I was a child because she was really afraid that I might drown." "...afraid that I drowned" means she thinks it already happened.

In addition to what the others say, I wouldn't normally use bathe in that way. I'd use the word swim for the beach.