Chances at either end were few and far between although Shrewsbury did strike the crossbar midway through the first half when midfielder David McAllister's deflected 25-yard drive looped up against the woodwork.
"The performance was full of anxiety and I believe we're better players than we've shown," said Shrews boss Turner.
"We didn't create an awful lot. We had a little flurry just after the substitutions and with a bit of luck David's effort might have gone in instead of hitting the top of the bar.
"But generally it was just a battle out there and to be fair to our players, Bury set up with a very difficult formation to play against."
Turner felt the tactics deployed by his opposite number Kevin Blackwell had much to do with his side failing to fire in the attacking third as they were involved in their third goalless stalemate in their last six games.
He added: "It was never going to be easy against them but I think we should have passed the ball a little bit better, more progressively at times. We were a little bit cautious on a number of occasions.
"Then late on in the game we had one or two opportunuties to shoot and didn't take the chance.
"It's always a concern when you're not scoring goals. That's two games where we've drawn a blank but at the same time we've kept a clean sheet and the side worked hard.
"We had two changes in the back four and they've come out of it okay. Of course, there's concern at the other end when you're not creating many chances and certainly in the first half there was very little created."