If you have a look at transactions, you can clearly see that all of them are done from SegWit addresses and SegWit was activated on 1st August, 2017, so, this should not be an issue because transaction fees become a problem since 2016 as far as I know.
The only way to tell whether the wallet is indeed making Segwit-enabled transactions is to see whether there is a witness data in the raw transaction hex.
That's true when we're talking about P2SH address (which has prefix 3...). But address input on all transaction which mentioned by @Synchronice use either P2WPKH (bc1q...) or P2TR (bc1p...) though.
But here it is different? Couldn't it also use an exchange so the transaction fee is a bit expensive?
Probably no. While exchange intentionally charge high withdraw fee towards it's user, they only spend fraction of it to pay Bitcoin on-chain transaction fee.