The Three Lions are winless in six games, their longest run since 1993, after a 3-3 thriller with Germany at Wembley on Monday.
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Southgate's side came back from 2-0 down to share the spoils, but there was plenty of discontent among supporters.
Harry Maguire made errors that led to goals, while Nick Pope's mistake late on allowed Germany to equalise.
This was the last match before the World Cup and there are plenty of concerns over how England will do in Qatar.
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Former England and Liverpool defender Carragher defended Southgate on social media after the Germany game.
He wrote on Twitter: "You would think @England had won numerous tournaments considering the stick Southgate is getting!
"It had been s***e for years before he got the job. Big Sam had the job before him!!"
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When questioned by a fan that he was scared to overhype a 'proper golden generation', he responded: "One or two players at the most from this current team would get into the 2002/06 team and the same with 1996/98."
Southgate will be heading into his third major tournament as England manager, with his record so far nothing short of brilliant.
He led the Three Lions to the semi-final of the 2018 World Cup before reaction the final of Euro 2020.
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"I've said all along, judge us on major tournaments. We've had two good ones where the fans have been incredible and pushed us the whole way," Southgate told talkSPORT.
"And we're going to need the same in Qatar, whether that's the fans travelling or the fans back home giving us motivation.
"You saw tonight they were incredible, giving us motivation. After going 2-0 down there weren't any boos or jeers, just pure determination.
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"Not just the players but everyone inside the stadium can go away proud as well.
"It shows the character in the team. We had a similar situation in the summer and kind of went the other way against Hungary.
"We spoke about not letting that happen again. The lads we have here, we've all got great mentalities, we've all got great pride wearing the England shirt and I thought you saw that come out in the last half hour."