following coins between blockchains
If you want to hold bitcoins anonymously then the only thing you can easily do is a cash transaction.  In the good old days you could just have mined some, but that's not an option available to most people today.

To answer the specific question, the exchange will have records of the receiving and sending transactions, so if you consider them to be compromised then there is no additional security available in using an exchange regardless of if you are moving currency within a blockchain or  between blockchains.