The Bulls saw their winless streak stretch to eight matches as the Londoners produced a performance that belied their recent problems.
With Bradford twice coughing up possession in the opening minutes, the hosts were quickly on the front foot with tries from Luke Dorn and Andy Ellis.
The Bulls seemed stuck for ideas, and things got worse for the visitors when Paul Sykes's kick bounced kindly for Dorn, the full-back speeding away and offloading for Jones-Bishop to cruise over the line.
Bradford threw themselves a lifeline midway through the half when, after neat footwork from Brett Kearney, Elliot Whitehead was on hand to finish.
Steve Menzies was penalised for dissent, Danny Orr booted the two points on offer and Bradford looked ragged, and after losing possession close to their own line Gale kicked a drop-goal on the hooter.
The Bulls began the second half in more determined mood and when Kopczak burst through Sykes was on his shoulder to score.
Suddenly the Bulls had wind in their sails and the hosts looked edgy, but all that changed when both Menzies and Michael Worrincy were penalised in quick succession, giving the home side the advantage which Jones-Bishop seized when he ran in his second try.
Menzies was sin-binned after yet another penalty, Orr successful with the boot, before Chris Melling rounded off a flowing move for the fifth try.
Steve Crossley added a late reply for the visitors but it was academic.
Harlequins: Dorn, O'Callaghan, Clubb, Jones-Bishop, Sharp, Orr, Gale, Ward, Ellis, McCarthy-Scarsbrook, Golden, Melling, Wilkes. Substitutes: Bolger, Bryan, Mills, Krasniqi.
Bradford: Kearney, Sheriffe, Platt, Nero, Halley, Addy, Sykes, Lynch, L'Estrange, Kopczak, Whitehead, Langley, Menzies. Substitutes: Godwin, Hall, Crossley, Worrincy.
Referee: R Silverwood (Dewsbury).