There is a difference between a "workout" like interval sessions and a "workout" like a tempo or just faster paced run.
Back to back interval sessions are generally not done. Interval sessions followed by a tempo day is done by some people.
It is hard to give an opinion on your coach's workout without knowing more.
Also it is hard to give advice without knowing your background.
But based on your question, I am guessing that you are a freshman in college, because I've heard questions like this from countless college freshman at this time of year over the past years. If so, keep in mind that you are probably running more quantity *and* quality than you have in the past, so you are pretty much guaranteed to feel terrible, even if your coach is assigning very appropriate training.
It is a very common story for freshman to do poorly, then bounce back with big PRs their sophomore year, because that first year they were struggling just to survive. By sophomore year, hopefully you have adapted and will start to show the benefit of the increased training load.