BTC transaction is slow now. How come when buying at Exchange, it's immediate? On Exchange, when buy or sell, it's immediate.
Isn't transaction is supposed to be ledger by Blockchain?
Does that mean my BTC account at Exchange are all not ledger yet. How legitimate is my BTC at Exchange then?
My understanding is when I withdraw BTC to my cold storage, that's the time it is ledger by Blockchain and it usually take some time. Am I correct?
When you purchase bitcoin from an exchange, it does not require an "on-chain" transaction.  They have a ledger, but it is not a blockchain.  It's a simple database, like the one that banks use.

Remember that when using an exchange, there is always someone on the other side of your buy or sell.  When you buy, that bitcoin comes from someone who transferred bitcoin to the exchange with the intention to sell it.  That transaction did require an on-chain transaction, but it occurred before you showed up to buy.  If you were to withdraw your bitcoin, that also requires an on-chain transaction from addresses the exchange controls, to addresses you control.

You are correct to question how "legitimate" bitcoin stored on an exchange is.  Firstly, because bitcoin you leave on an exchange is not your bitcoin.  It is the exchange's bitcoin that they are holding on your behalf.  Think of any bitcoin you keep on an exchange as an IOU from the exchange to you.  Secondly, exchanges have become insolvent in the past, meaning that they have more IOUs issued than they have either bitcoin or fiat to back up.  As a best practice, only keep as much fiat or bitcoin on an exchange as you require for your trading.  All else should be stored elsewhere.