While there is no actual law that I know of specifically stating that it is illegal for a teacher to prevent a student from using the bathroom, it is not that straightforward. A teacher, and the school in general, serves the roll of in loco parents. That is just a fancy way of saying that, when you are at school, your teachers are legally responsible for you the same way a parent would be at home.

This all means that if there were a safety or health reason that a student would need to go to the bathroom, your teacher would be legally at fault for preventing it. This does not mean that you can simply go to the bathroom whenever you like. It is still up to your teacher's judgement to decide if your bathroom trip is necessary.

I suppose that a note from a doctor stating that a student has a medical condition that requires frequent or regular bathroom breaks would legally enable a student to have unfettered access to the facilities. Other than that, it's still the teacher's call.

It is not illegal for a teacher to "not allow" a student to use the restroom.  A teacher must manage students and their learning and more than not a student can wait for the appropriate time for a restroom break.  There are certain situations and cases that there can be an exception. If a student has a documented health issue from a doctor, then a teacher must allow the student to use the restroom at their discretion. The health and safety of students is paramount and any teacher endangering a students health can be liable. Remember that many students just want to leave the room for undisclosed reasons and this can interfere with the class learning as well as the individual student!