My daughter has just graduated in a science subject and, as you say, this included having to write an essay every week for tutorial discussion. Some U/G subjects require more essays than this, but since she had also to attend lectures and lab work, this did represent a tough schedule.
Next month, she begins a one-year, taught Master's course, also at Oxford. Since she hasn't started yet, I have only the course handbook to go on. However, to my eyes, the workload appears terrifying! Each week seems to break down as follows: 12-14 hours formal contact hours (lectures etc), 15 hours for set/required reading, 8 hours for additional seminars and discussion groups, and then ongoing individual project work, leading to two project papers and a dissertation. There are also a variety of one-off field courses and study days every term. She will also have to allocate time for revision as there are examinations in each of the core modules.
But to answer your question, there are no weekly essays!