The midfielder joined on a four-year-deal from Motherwell last summer and was a rare shining light for the Hoops during a dismal campaign last season.
New manager Ange Postecoglou has had an indifferent start to life in the Parkhead hot seat, with Turnbull conceding recent results have not been good enough.
The 22-year-old, who will make his 50th appearance for Celtic in their Premier Sports Cup quarter final tie against Raith Rovers tonight, believes a winning run is needed to ease pressure on the squad.
He said: “We know Sunday wasn’t good enough, but the games are coming thick and fast.
“We had a meeting earlier this week to discuss everything and think about where we went wrong and how to put things right.
“We need to start implementing our style of play on every game, we have went over that and hopefully it starts clicking into place for us now.
“The supporters came out in their numbers on Sunday, and it was disappointing not to send them home happy.
“We have sent them home happy a few times this season and you feel good about yourself doing that.
“We need to put a good run together and I’m sure the fans will be behind us, they always are but it’s about providing the results for them.
“We need to go into every game with the same mentality. We know how good a team we can be, and we’ve showed that at times this season. It about doing that on a consistent basis.
“We played against a great side in Real Betis last Thursday and I felt we outplayed them for large parts of the game. There were just a few wee moments that killed us.
“On Sunday, we knew we weren’t good enough and know we can do better. I don’t feel we’ve been playing too badly but we need to up our performance levels.”
Turnbull admits the incentive of booking a semi-final place and a return to Hampden Park would give the club a welcome boost, but he acknowledges they must focus on beating Raith Rovers first.
He added: “No matter what opposition we face whether it be in the Europa League or a League Cup game against a Championship side, we all are fully focused on performing well and getting a positive result.
“We’re all looking forward to playing a good Raith side. We know they like to try and play out from the back, so we’ll need to be at our best to break them down and create goal-scoring opportunities.
“It’s a huge game. Everybody wants to perform at Hampden – it’s one of the biggest stages in Scottish football.
“Going to the national stadium, you always know you’re in an important game and getting to a semi-final would be great for us.
“Of course, it would be great to get some silverware so early in the season to give everybody confidence going into the second half of the season.”