The Ashes has already been decided with Australia taking the first three Tests, but England is scrambling to avoid an embarrassing 5-0 series whitewash.
Bairstow said captain Root was coping with the growing pressure and the England team was working hard to break their run of outs against the Australians in the fourth Test, starting at the Melbourne Cricket Ground on Tuesday.
"We don't just owe him (Root), we owe ourselves as well," Bairstow told reporters.
"So without doubt in these next two games it's a case of, in our eyes, a two-Test series now and that's exactly how we've got to look at it.
"We've got to go out there and try and win both of these Test matches."
Root has been in the firing line leading a disastrous England tour and has been blasted by former Australia captain Ricky Ponting as looking like a "little boy" who has been a "bit soft" during the Ashes defeat.
"Those things show a complete lack of respect for him as captain," Ponting told reporters.
"To be honest, I think he has been under pressure right from the start because of things that have been happening off the field."
Bairstow "headbutted" Australia's Cameron Bancroft in a bar in October, while this month England Lions batsman Ben Duckett was suspended for pouring a drink over England bowler James Anderson.
Ponting said England need Root to "step up big time" and criticised him for being "too shy" in his post-match news conferences.
"You need to be more than that as a leader, especially when things aren't going well. It just looks like it has been a little bit soft."
- England 'blown away' -
========================
Ponting, who captained Australia in 77 of his 168 Tests between 1995 and 2012, said England had been "blown away" in the series.
Despite Root winning all three tosses, Australia won by 10 wickets in Brisbane, 120 runs in Adelaide and an innings and 41 runs in Perth.
Ponting said Root's batting -- he has scored only 176 runs at 29.33 -- was showing the effects of leading a beaten team.
"Look at some of the shots he's played. It looks as though something is playing on his mind a little bit," Ponting said.
"They are not the shots you would expect from one of the best players in the world, and certainly not someone trying to lead from the front."
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