Schumacher headed home Gregg Wylde's corner in the final minute to clinch a point at the end of an entertaining encounter at Gigg Lane.
Aaron Wildig put Shrewsbury in front inside the opening minute but Bury quickly hit back when Matt Doherty slotted a low shot under goalkeeper Chris Weale.
The Shrews edged back ahead when Jon Taylor raced down the right to finish well after 68 minutes, but Schumacher's goal ensured the points were shared.
Turner said: "I'm ever so frustrated because I see us passing the ball for fun at times, particularly in training, but in the last 20 minutes of games we keep inviting pressure by giving the ball away.
"We know they're chasing the game and they're going to leave men up front so you have to match them for that.
"There were so many times we could've got out with the ball and passed it with some precision, which we're good at, but we let ourselves down. I've never known a side who give the ball away so frequently when we just need to take the sting out of the game.
"We keep passing the ball to the opposition, which baffles me because we've got good passers in there. That's our strength.
"I've got to say, maybe we would've settled for a point before the game in the circumstances but we've got to start winning our away matches. Never was there a better chance than here."