Conway survived a vociferous lbw shout on an umpires call before he had opened his account as Taskin Ahmed and Shoriful Islam bowled testing lines and extracted good movement through the air and off the pitch.
After the first hour, New Zealand had only 15 runs on the board from 13 overs before Conway and Young took control.
Conway, in his first full international since breaking his hand during the T20I World Cup, brought up his 50 pulling Mehidy Hasan high over the boundary.
With that, he became the first New Zealander to pass the half-century mark four times in his first seven Test innings.
The left-hander, almost assured of taking over the number four slot when Taylor bows out of Test cricket at the end of this series, smacked two more deliveries to the boundary in the same over.
His 88 at tea included 13 fours as well as the six.
Bangladesh missed a chance to remove Young in the first over after lunch when a Shoriful delivery swung away late and was well taken by an athletic Liton Das.
There was no appeal by Bangladesh, who thought the ball had beaten the bat, but Ultra Edge technology indicated the ball shaved the outside edge and had there been an appeal, Young would have been out for 27.
He was to add another 25 runs before being run out when a pinpoint throw to wicketkeeper Das saw the first bail lift off the wicket just as his bat reached the crease line.