BTC transaction is slow now. How come when buying at Exchange, it's immediate?

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?

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