When swapping coins for ETH, would the network fee be the same for that transaction no matter which exchange/swapping app/wallet you used?

For instance, if I wanted to swap 1 BTC for 13 ETH, would the network fee be $500 USD (for instance) no matter where I made the swap?

I'm swapping an alt coin for ETH using the crypto.com wallet and the network fee is making me sick to my stomach…