The visitors ended the day on 176 for six, still miles away from the daunting 509-run target.
Taijul immediately put Bangladesh in command, trapping Ireland captain Andy Balbirnie for 13 and then removing opener Paul Stirling for nine in successive overs to rock the top order.
Hasan Murad then made further inroads, dismissing Cade Carmichael for 19 after his 51-run stand with Harry Tector, who held firm for a gritty half-century before falling to Murad attempting a big shot.
Khaled Ahmed added to Ireland's troubles by sending back Lorcan Tucker for seven, while Taijul later removed Stephen Doheny for 15 to keep the visitors under constant pressure.
Curtis Campher (34*) and Andy McBrine (11*) held on till stumps but face a daunting final day.
With the highest-ever successful Test chase being 418, Ireland now confront a near-impossible task of either rewriting record books or surviving an entire day to force a draw.
Earlier, Bangladesh strengthened their position as Mushfiqur Rahim, in his 100th Test, followed his first-innings hundred with an unbeaten 53, guiding the Tigers to 297 for four before the declaration.
He shared a 123-run stand with Mominul Haque, who fell for 87 soon after lunch.
Bangladesh had resumed the morning on 156 for one and dominated the early session, adding 124 runs as Shadman Islam fell for 78 and captain Najmul Hossain Shanto departed for one.
Ireland also missed a key opportunity when Carmichael dropped Mushfiqur on 16, a chance that proved costly.
