Centralized exchanges work with companies that track cryptocurrency transactions which creates balance between keeping transactions private and following rules. Companies like Chainalysis are good at tracing where money comes from and goes to.
First and foremost, they are strict in Terms of Service against their customers for aiming at protecting their business as most as possible, with AML/KYC policy. Centralized exchanges nowadays are mostly enforced mandatory KYC and customers have to reveal their individual identity documents before using those accounts.
If you send your cryptocurrency through another address it might not completely hide your tracks but it can give you some protection. You could say money came from friend or as payment for something which might not fool everyone but makes it bit harder to figure out where money really came from. This method does not make you completely anonymous but it does make it more complicated for others to track your transactions.
Firstly you have to avoid using mixers/ tumblers from your side. Avoid using Coinjoin transactions by yourself.
Secondly, if you receive bitcoin or cryptocurrency from another source such as from any company or manager, you must receive it in your non-custodial wallet first, and move it to another wallet or 2 other wallets, before deposit it to your account on centralized exchange if you actually need to send it to CEX.
This video can help you understanding more about wallet fingerprint and how exchanges detect tainted coins.
Bitcoin Q&A: Blacklists, Taint, and Wallet Fingerprinting