Your teacher is right. Listening to English songs is not a good way to learn grammar. Many native speakers do not learn grammar and do not know grammar as a subject, but what they say and write is most of the time still grammatically correct. That is because they grow up in an English-speaking environment. Native speakers who wish to learn grammar use a grammar textbook. That is what you should do as well. You need something on top of that: you need to attend classes run by a qualified teacher.

Maybe I'm a rebel! - I don't think you need to exclude listening to music because there are some songs that will help you learn grammar as well as vocabulary. Just keep in mind that not all music lyrics are going to be used as a "natural" way of speaking in conversational settings. I would advise you to play games on this website that uses music videos: http://lyricstraining.com/ You have to click on the words in the gaps This website has a list of lyrics: http://lingual.net/category/music/ Click on "check the script" while you are listening. Listening is an important part of learning how to have conversations.