England are feeling Euro pressure after drab Denmark draw: Southgate
Southgate's side arrived in Germany as the bookmakers' leading contenders to win the tournament.
But they have failed to match those lofty expectations in their first two Group C matches.
England were fortunate to escape with a 1-0 win against Serbia after a spluttering display in the second half of their opener.
And they produced an even more limp performance in Frankfurt on Thursday as Harry Kane's 18th minute strike was cancelled out by Morten Hjulmand's brilliant long-range blast before the interval.
"Clearly we are disappointed with the level of the two performances. We have to analyse that in depth and find some solutions to the issues we have. We will spend a lot of time doing that," Southgate said.
"We know the level can be higher. Maybe the biggest thing is we have to accept the environment we are in and walk towards the expectations.
"We are in the environment of winning on the biggest possible stage. These boys aren't lacking effort. We have to find more quality in what we do. If anything they are showing they care too much."
England would have wrapped up first place in the group with a win over the Danes, but instead were greeted with jeers by frustrated fans after their lethargic display.
"If we don't win we have to accept what comes our way. I can completely understand the fans' frustration with the way we played," Southgate said.
"That is my responsibility as the manager. I have to find solutions. England have never won back to back group matches I was told. There is a reason for that. We have to stay calm inside the group and find answers.
"The challenge is to do something that has never been done before. So it couldn't be any bigger."
'An anxious performance'
Southgate said the key to England's failings against a combative and intelligent Danish side was their failure to press hard enough and a lack of quality on the ball.
"We have to accept that we didn't press with enough intensity. We kept conceding possession too easily. When you do those things it's hard to have control, it led to an anxious performance," he said.
"We have to be better if we are to progress to the later stages of the tournament and deliver what everyone expects us to do."
Southgate, whose side sit top of the group and remain on course for the last 16 despite their flaws, substituted Kane in the second half despite his goal in a rare decision to replace his captain.
"We felt the whole front line put a lot of work into the game. Harry has only had one 90 minutes in the last few weeks. The game last week took a lot out of him and we wanted to get more speed into the front line," Southgate said.
Kane's lack of match fitness, after injury ended his Bayern campaign prematurely, is a recurring theme for Southgate, who is also without first-choice left-back Luke Shaw, while Arsenal winger Bukayo Saka is working his way back to form after a fitness issue.
"We don't feel the way to press is really high up the pitch," said Southgate, whose side face Slovenia in their last group game on Tuesday.
"We know we have had a lot of issues in the lead up. I don't think that's the physical level of the team right now.
"But we have to find a way to be harder to play against than we have been in the last three halves."