Once they fell within six overs of each other, England were quickly finished off by a ruthless South Africa side.
Jonathan AgnewBBC cricket correspondent
"You must sell your wicket dear when you are trying to save a Test. England need to be more switched on at Headingley because, ultimately, their success or failure will come down to the application of their batsmen."
Steyn did the damage on Monday, removing Ravi Bopara half an hour into the day before dismissing Bell, Stuart Broad and Graeme Swann to give him seven wickets in the match.
With Bell and Prior at the crease, confidence grew that England could achieve the unlikely and save the Test, the first in a three-match series that will decide the best team in the world.
Bell was particularly stoic, reining in any attacking instincts on his way to a hard-working half-century.
His steadying influence helped England recover from the early blow of losing the under-pressure Bopara, who followed up his first-innings duck by dragging a back-footed force off Steyn on to his stumps.
The ugly nature of the dismissal seemed to focus Bell and Prior on the task at hand and they knuckled down for the remainder of the morning session, taking the sting out of the seamers and using their feet to negate any turn from leg-spinner Tahir.
Bell did get a faint edge on one Tahir delivery, but stand-in wicketkeeper AB de Villiers spilled the chance, helping England reach lunch without further alarm.
However, Prior's departure sparked a collapse that saw England lose their last five wickets for 37 runs.
The sweep shot was redundant until Prior attempted it against Tahir, succeeding only in gloving the ball to Jacques Kallis at first slip.
The importance of the wicket was not lost on South Africa, who took the new ball four overs later and set about finishing off England's tail.
Steyn, bowling with great hostility, drew Bell into playing at a wide one that found Kallis at second slip.
England v South Africa
First Test, The Oval - South Africa won by an innings and 12 runs
Second Test, Headingley
Third Test, Lord's
Broad was caught down the leg side after South Africa successfully reviewed a not-out decision, Swann drove tamely to extra-cover and James Anderson was trapped in front by Tahir.
The second match of the series starts at Headingley on 2 August, with South Africa knowing a victory will see them overtake England as the world's leading Test team.
Listen to analysis of each day's play with Jonathan Agnew and Geoffrey Boycott on the
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