Gammatical problem

Most of the patients who took the new medicine got well after one week. – correct.

First of all, you have to identify the relative clause – “who took the new medicine”.
A relative clause adds extra information about one of the nouns in the main clause.
The main clause in the sentence is “Most of the patients got well after one week”.

There are two types of relative clause ~ defining relative clause and non-defining clause.
Defining - to give essential information in order to identify what/who you are talking about.
Non-defining – to give extra information, not essential for identifying what/who you are talking about.

In this case, “who took the new medicine” is a defining relative clause. We don’t use a pair of commas to separate the defining relative clause from the rest of the sentence. A defining relative clause is essential to clear understanding of the noun,

It is important to identify what patients – the ones who took the new medicine.

Consider another sentence.

Most of the patients, who lived in Kowloon City, got well after one week.

“who lived in Kowloon City” is a non-defining relative clause. This relative clause adds extra information about the subject. We usually use commas to separate the non-defining relative clause from the main clause.

I think your teacher is correct since the sentence is referring to the patients who took the new medicine, but not other patients. Therefore, the clause 'who took the new medicine' should be a defining clause instead of a non-defining clause since it is not extra information.