You can use Bitstamp or BTC-e. Both are good and secured.
I have all my fiat in a EURO Flex account bitcoin address as Fiat. How can I move them to Bitstamp?
Buy bitcoin there and move BTC then sell in other exchange to keep in fiat.
I am afraid I will lose many EUR in fees
I think you must convert all of fiat money (EUR) in bitcoin then withdrawal all of bitcoin to offline wallet bitcoin,
so you will not lose money because of fees because in bitcoin you can be full of control your money
and put money in bitcoin there is not inflation like in fiat money.