South Africa were 525 for six at tea, 104 runs behind England's first innings total of 629 for six declared.
Stuart Broad and James Anderson struck three times with the third new ball to cause some short-lived alarm for South Africa but Bavuma (52 not out) raced to his second Test half-century off just 52 balls, with 11 fours, as he and Chris Morris (24 not out) shared a bright unbeaten partnership of 76 for the seventh wicket.
The third new ball was taken immediately after lunch and Amla added only one run before he was bowled for 201 by Broad, ending a stand of 171 with Amla. Four balls later Du Plessis was caught at gully off Anderson for 86.
Quinton de Kock made only five before he was caught at backward square leg off Broad, bringing Bavuma and Morris together with their side still 180 runs behind.
But Bavuma and Morris counter-attacked, posting the fastest fifty partnership of the innings off only 48 balls.
England's catching woes continued when Amla was dropped by James Taylor at short leg off Broad shortly before he was dismissed, while Morris gave a caught and bowled chance to Steven Finn when he was on 22.
Amla survived three chances before reaching his double century, while AB de Villiers and Du Plessis, who both shared big partnerships with the South Africa captain, were also dropped.
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