Binance is the perfect exchange for what you're looking for, it offers both anonymity and a high volume. Besides Binance there is Cryptopia, which also has a lot of different coins listed and doesn't need verification, the only downside is it's volume.
Binance is the perfect exchange for what you're looking for, just scratch that out. There is a compulsory verification at Binance before you are allowed to make a withdrawal. But you can trade without making the verification at the beginning of opening an account with Binance. The verification is done on the Binance site on PC or on a laptop, not phone.