England had resumed at the Shere Bangla National Stadium in Mirpur on 171-3 with Warwickshire wicket keeper Jon Trott on 64 and Bell 25 not out.
Trott was out without adding to his total, bowled by Shakib Al Hasan but Bell hit some superb shots, hitting fifteen fours and a six from 262 before he was out with England's score on 415-6, caught by Jahurul Islam off the bowling of Shakib Al Hasan.
Tim Bresnan, who finished the day on 74 not out helped England past Bangladesh's first innings total of 419 and they closed 21 runs ahead with two wickets remaining.