Clarke, whose poor form with the bat continued with scores of 10 and 3 in Birmingham, grassed a regulation catch at second slip.
The ball flew through what is considered one of the safest hands in the team when Bell was on 20 and England were 1-35.
It won't help the 34-year-old dispel suggestions the end of his international career is nigh.
More importantly for a skipper trying to guide Australia to their first Ashes win in England since 2001, it helped the hosts cruise to a 2-1 lead in the five-Test series.
Clarke has a mountain of work to do before the fourth Test starts in Nottingham on Thursday.
To find a way out of a personal slump that suggests this series could be his last. To find a way to pick up a side that was outplayed for the most part of a helter-skelter match.
To find a way to win an Ashes series from 2-1 down, a feat achieved only once - by Don Bradman's team in 1936-37.
"Our goal is to win the series and I still believe we can do that," Clarke said. "Very hard to explain it (playing so poorly after a 405-run win at Lord's)." Changes to the XI are likely.
The visitors had their issues with the ball, but far more alarming were woeful collapses of 5-60 in the first dig and 4-30 in the second innings.
Clarke, Adam Voges and Mitch Marsh combined for just nine runs on Thursday, when Steve Finn ran rampant en route to career-best figures of 6-79 in the second dig. It's been 43 years since Australia have lost five wickets for 100 runs or less in both innings of an Ashes Test.
Nevill and Starc shared a stand of 64 runs on Friday. Both men faced over 100 balls - more than any of their teammates in either innings.
"It just shows you how badly they played in the first two days," Shane Warne said on Sky Sports.
"They didn't bat time and let the pitch dry out. It would have been a good Test match."
The wagging tail was also a potential portent of how England could struggle in the fourth Test without Jimmy Anderson. Anderson's side strain and a score of 12 from out-of-sorts opener Adam Lyth, who unsuccessfully reviewed his lbw verdict, are the only concerns for Cook.
Bell stroked five boundaries from his first nine balls, finishing 65 not out to make it consecutive half-centuries in the game. Recalled paceman Finn, who dismissed Clarke and Steve Smith twice, was named man of the match for his eight wickets in the match.