Two goals inside nine second-half minutes from Blades captain Michael Doyle and Dave Kitson, a cracking 25-yard strike, proved decisive.
Shrews, now just four points clear of the relegation zone, pulled a late goal back as Stephen McGinn's long-range free-kick seemed to elude everyone on its way in, but they were unable to prevent a first home defeat since October.
The result came on a day in which Turner's team were without injured striker Marvin Morgan and then lost top scorer Jon Taylor to a groin injury at half-time.
"We were beaten by a side who were stronger than us, better than us," admitted Turner. "They had an element of luck about the first goal but we weren't tight enough for the second which was a good strike.
"Overall we've got to be disappointed with our own performance but they're a decent side and you have to sometimes hold your hands up.
"We've worked hard to defend but going forward we didn't produce anywhere near enough."
Turner added it was important his players now retained their self-belief despite the gap between themselves and the bottom four being cut.
"The teams below us are capable of getting wins, just as we're capable," he said. "There's concern in there, that's obvious, but we've still got some cushion.
"It wasn't so long ago we were in the bottom four and we've eased ourselves out of that. That's only our third defeat in 15 games so players need to maintain that belief. What they've got to do is scrap for everything that's put in front of us and we'll have to do that next week at Tranmere."