England's failure has led to questions over Root's position, particularly given his inconsistent batting throughout the series.
But Australian Bayliss, who leaves his role after four years following this week's final Test at the Oval, said there were no issues with his skipper.
"He's not come under question from anyone making any decisions," Bayliss said.
"He's under no pressure at all." Root, sixth in the world rankings, averages less than 31 in the series so far and has been dismissed for nought three times.
Speaking about Root's form with the bat, Bayliss added: "Everyone goes through periods where they don't score as many runs as they'd like.
"I think the Australian team have bowled pretty well to him, so we'll see how things go further down the line.
"From my point of view, he was our premier batter and the Australians always try and target the opposition captain. He's played well when he's got starts and I don't see too many problems."
After winning at Old Trafford to go 2-1 up in the series, Australia's players celebrated on the field and were accused by some of mocking England spinner Jack Leach while doing so.
"Fake news, like Donald Trump says," said Bayliss.
"No one has mentioned it or spoken about it. It's all talk from you guys (the media)."
Bayliss will return to Australia after the series, having spent four years in the job and taking England to World Cup glory in July.
"On one hand I am sad to be leaving," he said. "It's been a fantastic environment to work in, everyone has made me feel at home.
"From a cricket sense we've had a lot of good performances, the cherry on top the World Cup win.
"But I am looking forward to getting home and it's time for someone else to come in with a new voice."
Anderson Backs Root
Injured England paceman James Anderson said Root remained the best man to captain the side and would have the chance to forge a relationship with Bayliss's successor.
"He has done some great work," he said.
"But the team has been developing over the last few years and going into a big series like this you want a settled team that's fit and firing. I feel for Joe as he's not quite had that. It's been difficult."
Anderson, England's all-time leading Test wicket-taker, said he had no intention of retiring despite an injury-ravaged season.
"One of the big frustrations about not playing is seeing the guys out on the field," he said.
"It's been great to see Jofra (Archer) taking to international cricket the way he has and Broady (Stuart Broad) has been fantastic all series. He's shown his class all the way through.
"I'd love to be a part of this bowling attack because I think we could have something special there."
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