England reached stumps on 78 for four in their second innings, still 56 runs from making India bat again, after the early losses of captain Alastair Cook, Moeen Ali, Bairstow and Ben Stokes.
To make matters worse the tourists fear Haseeb Hameed could be ruled out of the remainder of this series after the injury to the teenage opener’s left hand prevented him from batting on Monday.
Hameed was struck on the little finger of his left hand by Umesh Yadav on the first day of this match on Saturday and although he has batted in the nets since he was unable to take the field on day three.
England will give Hameed a scan after this Test has finished. But even if he bats on Tuesday, the 19-year-old may be ruled out of the final two of the series in Mumbai and Chennai if the scan shows a fracture.
England are likely to be 2-0 down by the time the fourth Test starts on Thursday week after they conceded a 134-run advantage by allowing India to post 417 in their first-innings and then losing four top-order wickets before the close.
Bairstow admitted: “It’s not the day we were looking for, to be brutally honest. Me and Ben getting out at the end there caps off a bad one for us. We thought this morning, if we got a couple of quick wickets, we were in a good position. That didn’t happen and we’re now fighting an uphill battle.
“We’ve got to take it on the chin and fight hard tomorrow – we’ve got a lot of batting still to come, we must remember that.”
Bairstow at least reached a significant personal landmark on the third day of this match, his catch of Umesh Yadav that wrapped up India’s first innings seeing him break the record for the most Test dismissals by a wicketkeeper in a calendar year
Bairstow’s tally of 68 overtook the previous best of 67 held jointly by Australia’s Ian Healy and South Africa’s Mark Boucher.
He said: "There was a lot of speculation a year ago about whether or not I should be keeping wicket, so to put the hard work in and have that come to fruition is a very special achievement for myself. "It’s still not anywhere near where I want my keeping to be and I’m not under any illusions. You look at yourself and you have two options: you hide in a corner or you front up and take it on the chin. Thankfully I took the latter option and it’s starting to pay off."