The Shakers put in a battling performance as they aimed to bounce back from successive home defeats.
But Bradford sub Conlon dealt Alan Knill’s men a cruel blow as he headed home in the 86th minute.
The game had started slowly with the first shot not coming until the 12th minute, when Bradford’s Michael Boulding fired narrowly wide. The Shakers had a sight of goal when Bradford goalkeeper Rhys Evans mis-hit a clearance straight at Elliott Bennett, although his 40-yard effort drifted well wide.
Bury then breathed a sigh of relief after Omar Daley burst down the left, leaving right-back Paul Scott for dead.
His cut-back was touched on by Boulding and although Nicky Law had time to pick his spot, the Bradford midfielder scuffed his effort wide.
Bury replied in the 29th minute with Dawson leading a break forward and curling a 20-yard effort against a post from Mike Jones’ lay-off.
The hosts finished the half on top, with Law having a shot blocked by Brian Barry-Murphy before firing wide from the edge of the box. Dawson curled a difficult chance over shortly after the restart but the Shakers almost gifted Bradford the lead moments later.
Ryan Cresswell’s back-pass was wide of Wayne Brown and the Bury keeper had to scamper back to claw the ball off the line to prevent an own goal.
Cresswell redeemed himself by getting back to prevent Thorne getting a shot away and Bury soon regrouped.
Barry-Murphy’s 61st-minute free-kick found Efe Sodje at the far post and his header forced Evans to tip over.
The game began to open up, with Brown doing well to deny Thorne while Andy Bishop tested Evans.
With the home crowd urging their side forward, Bury began to look dangerous on the break.
But Conlon struck four minutes from time, rising above Brown to head in the rebound after Graeme Lee’s long-range free-kick struck the crossbar.
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