England paceman Ben Stokes dismissed Rahim three overs before the stumps after the Bangladesh captain looked to have taken his side to a comfortable position with a solid innings of 48 runs.
Bangladesh survived some anxious moments after Rahim's dismissal to finish the second day with 221-5, trailing England's first innings total of 293 runs by 72 with five wickets in hand.
Shakib Al Hasan, who shared 58 runs with Rahim for the fifth wicket, remained unbeaten on 31 overnight along with nightwatchman Shafiul Islam, who was yet to open his account.
Tamim, who hit seven fours, added 90 runs with Mahmudullah (38) for the third wicket to steady Bangladesh's innings after Moeen Ali took two wickets in one over before the lunch break to rattle them.
Tamim got a reprieve at 55 through a television review after being given out just before the tea break.
Adil Rashid, who earlier dropped a tough catch of Tamim at 28 off Moeen, broke the left-handers' partnership when he had Mahmudullah caught at slip by Joe Root.
Earlier, 18-year-old debutant off-spinner Mehedi Hasan claimed the final wicket of Stuart Broad to finish with 6-80 as England added just 35 runs to their overnight 258-7 before they were all out.
Left-arm Taijul Islam took the other two wickets that fell in the morning in England's innings to return with the figures of 2-47.
Taijul dismissed overnight batsman Chris Woakes (36) off the first ball of the day as Mominul took the catch at forward short leg.
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