Title says it all. Is there a bitcoin exchange that pays most of fees for you? For example, I want to send Bitcoin to my Electrum wallet but others want to withdraw too. So like my transaction is bundled with many other people's transactions.
You and others have to withdraw your bitcoins from that centralized exchange. You can not send your bitcoins from your exchange account to Electrum wallet. You don't have rights per Terms of service and don't have private key to make any transaction on chain. You only can submit your withdrawal request and wait the exchange to approve that request, and proceed the transaction for you.
Basically, people willing to pay higher fees help to substantiate the poor man's low fee in terms of how quickly one wants receive Bitcoin. I am willing to wait even 2 days before I receive Bitcoin. Due to high on-chain fees, I cannot manage fees.
You can not manage fee because you don't have private key with your exchange account.
Exchanges are greed and they don't care that transaction fee on Bitcoin network is expensive or cheap than usual. They charge you a fixed withdrawal fee per one withdrawal. The joke is when transaction fee is cheap, they don't change it but if transaction fee goes high, more expensive, they will increase that fixed withdrawal fee and charge more fee on users.