It is impossible to have a trading sites that do not charge a fee in doing transactions like buy and sell and also in doing withdrawals. Fees in trading site is one of the things that make the sites strong, because fees serves as an income to them which they use it to maintain the site and use it as a fund in doing other type of progress in their site like improving their site or having a new feature. Fees is always required and it's impossible to have a site that does not charge.
Impossible? No it is not, If I remember it correctly there is an exchange site from Indonesia named vip.bitcoin.co.id that does not charge any fees on trading btc to altcoins vice versa but they only charges fees on btc-fiat trades. The bad thing is that there is only few altcoins available on the markets (less than 10alts) and of course the rate is lower than other trading sites like bittrex/poloniex.
The fees at every transaction are needed to pay up the costs of trades execution, servers costs, and to pay salaries to cryptocurrency exchange team.
Exchange without fees would have to pay for their customers to give them an option of selling and buying different cryptos. Good samaritans like that does not exist.
Every single company has to generate profit, to be working.
If it is not giving profits but losses, there are two ways- change the weak parts in the business plan, or just give up and slowly bankrupt.
You cannot expect getting something for free, you need to get used to paying fees: both for blockchain transactions, and exchange operations.