Most of the Bitcoin exchanges, are they using 3rd party API like blockchain.info or running their own node? I'd like to know how the bitcoin exchanges work. It's not about big exchanges like binance or bittrex. I'd like to know people could make an exchange solely relay on third party apis and are they safe ?
Most of the Bitcoin exchanges, are they using 3rd party API like
  blockchain.info or running their own node?


They're running own nodes.


  I'd like to know people could make an exchange solely relay on third
  party apis and are they safe ?


In my own opinion it's not safe to delegate service to third party services in case of money, especially if you can create and run that's service byself.