Wales Under-21 international Bradshaw had not scored a league goal since October 2011 when he scored against Accrington.
But he brought that near two-year wait to an end when he came off the bench and then robbed defender Nathan Byrne before drilling home five minutes from time.
It ensured Shrewsbury, much the better side, finally doubled their advantage after impressive debutant Adam Reach, on loan from Middlesbrough, had fired them into an early lead.
"It might sound daft but I think it just took the pressure off Tom, leaving him out," said manager Turner.
"He's played the first three games and I think I'm right, he had barely had a shot at goal, but coming on as sub when he's fresh and one or two are getting tired, that's his strength, his mobility and his pace. He got himself free and might have got more from that position."
Turner was relieved to see Bradshaw's strike hit the net after admitting he was a little concerned with the number of chances his side squandered at 1-0 ahead.
"You fear the worst when you're throwing away chances like we had, but we stood strong at the back," he said. "We were solid as a team and I thought we deserved more than the two goals."
Turner was also pleased with the impact of Reach, his latest loan recruit, who offered Shrewsbury an extra dimension wide on the left.
He added: "Adam did well on his debut, it was a good goal and he might have had another one. I think he can score goals and I think he's scored goals on his debuts wherever he's been."