How can I say "a class in Chinese" (and not "a Chinese class")?
The succinct way to express this can be: 中文授课. Put it into a sentence: 本课程用中文授课。This curriculum was taught in Chinese. 

This is a class in Chinese can be: 这门课用中文讲述 or 这门课用中文授课.


  New students need to sit the HSK4 exam, because the teachers use Chinese to teach their classes.


It can be translated to 新学生需要参加ＨＳＫ４考试，因为老师(们)用汉语授课。their can be implied in this case because every body would know it's their classes. 

Colloquially, you can replace 授课 with 教课 or 上课. So, 汉语教课 and 汉语上课 are also fit and succinct.