Growth has never been the hard part for music startups. Making money has.
The reason turntable is popular is due to the social elements, specifically vanity. You're on a stage in front of an audience getting points for how sick your tracks are. It's a basic human emotion that is triggered that makes the site so popular. The discovery aspect is great as well but I don't think it's as important to the sites success.
I do think they have a huge legal problem looming, especially with regards to letting you play tracks others have uploaded. Google and Amazon are already fighting the 'public performance of copyrighted works' issue, and turntable is far more blatant.
Definitely agree on the legal problem here. The whole non-US shutdown doesn't look promising.