That doesn’t sound strict at all. That sounds like a standard expectation of behavior. Be on time or it will be counted as late. How is that not fair? It is a literal description of the situation. Go to the bathroom before class. College students are adults. The typical college student has had at least 18 years of experience in regulating their bladder. Stop at the bathroom before class. A four page essay is pretty much standard fare in an English composition class. If this is Comp I, then it’s not at all unusual. When I took English Composition I, we had 5 papers throughout the semester, each of which had to be 3-4 pages in length.
Personally, I like when professors have these policies. When students walk in late or get up in the middle of class to use the bathroom, it is distracting.