Most online wallets (including exchanges) do not provide private keys and only provide an address.
If you can’t access your private key, you do not have a wallet.
There's been several exchanges which have been targeted by hackers in the past.
I think you confuse exchanges sites and web wallets. "do not trust both to
Store your coins
As a result this is costing you money or time. It's a pretty well known fact that Blockchain.info in particular is often wrong in it's estimation in fees.
You are not obliged to abide by those fees
Let's take a look at the well known inputs.io website which was ran by TradeFortress.
When you want to store your coins, make sure you have access to your private key and then choose a well-programmed wallet from ---->
https://bitcoin.org/en/wallets/Warning: online wallets are banks that you are trusting