Mackay was sacked on Friday following an ongoing dispute with club owner Vincent Tan.
City went on to draw 2-2 with Sunderland on Saturday but let a two-goal lead slip in the closing stages and Turner insists they must retain their focus if they are to remain in the Premier League.
"We have to move on. If we don't then we will get relegated. We have to look forward. It's what we want to do," he said.
"That's the one thing the old manager would say, once the game is over look to the next one.
"Whatever's happened, you focus on the next challenge. So we want to look forward and he'd have wanted us to carry on.
"We are at the halfway point (of the season) and we have a good chance of staying up."
Turner admits they have not had time to fully come to terms with Mackay's departure but believes they are responding in the right way.
"We have not had time to digest it yet because with the Christmas fixtures they are coming thick and fast and we have had to concentrate on the game," he added.
"I think we approached the Sunderland game in the right manner, as we would have regardless of what's been going on around the club.
"Maybe now we will have time to digest what's actually happened over the last few weeks and then we move on to Arsenal away.
"The manager brought me here and showed a lot of faith in me, but at the same time nothing is forever in football and you have to look forward."