Another defeat looked a possibility as Hull fell 12-0 behind to tries from Paul Sykes and Royston, Patrick Ah Van converting both.
But the home side, without the suspended Sean Long, battled back to level as Kirk Yeaman crashed onto Willie Manu's offload to score, before Danny Tickle, who kicked five conversions, took three tacklers over the line with him after taking Sam Obst's pass.
Chev Walker had the Bulls in front again following Marc Herbert's flick-reverse pass, but Hull were on terms once more when Lee Radford dived over.
Turner crossed four minutes after the restart to give Hull the lead for the first time.
But it was the Bulls' turn to fight back when, despite being reduced to 12 men when Ian Sibbit was sin-binned for a grapple tackle, Royston took advantage of Will Sharp's lapse of concentration to nip in and touch down Brett Kearney's kick.
Herbert then missed a drop-goal attempt for Bradford and was made to pay when Briscoe went in at the corner off a Yeaman pass.
And, with time running out, Briscoe wrapped things up when he nipped in for his match-winning score.
Hull FC coach Richard Agar: "I thought we had a pretty good performance for most parts.
"I thought Bradford had been defending really well and in the first half we threw plenty at them, without quite finishing at times.
"In the second half we would have been better advised at times to plough our way down the field and turn the ball over, but we did enough to get a couple of late tries.
"I think over the 80 minutes we were good enough with the ball and created enough."
Bradford coach Mick Potter: "We came trying to get two points. We put a lot of effort in as we knew we'd come up against a tough team and we weren't quite good enough.
"We came up with some errors we shouldn't have done, especially in our own end. We held them out for six sets in a row, so our defence is improving.
"They came up with a jammy kick that bounced around and they scored. That was a real backbreaker for us."
Bookmark with:
What are these?