The £27million defender knows he has struggled to adapt since his summer move from Southampton.
New United boss Louis van Gaal criticised his fitness levels in the summer and gave him his own training plan - something Shaw admitted he was right to do.
And the 19-year-old, who played in United's 2-2 draw at West Brom on Monday, has given an honest assessment of his start at Old Trafford.
He said: "They haven't seen the best of me yet
I know I haven't been up to my full potential and I think maybe people can see that at the moment
I am trying hard and it's about getting used to everything.
"When you come to a club like this it's never easy for a young player like me
But I'll get myself started soon and put in good performances.
"I wouldn't say I've been horrendous but I know I can do much better
And I need to do better
I've come to Manchester United and I know what they need
I've been working hard and the performances will come."
Shaw, who played for England in this summer's World Cup, also dropped back to the Under-21s this month to help them qualify for Euro 2015.
But he felt playing two games against Croatia was vital to boost his fitness.
"It always helps and the main thing for me was it was two 90 minutes for fitness
I feel fit but like I said I haven't played my best recently," said the left-back.
"That's football
Hopefully I'll come out of it now
I think I could still get a bit fitter from working hard in training
But I do feel pretty fit in training."
United host Chelsea in the Barclays Premier League on Sunday already 10 points behind Jose Mourinho's leaders.
Chelsea are unbeaten in the league, winning seven of their first eight games and are five points clear, but Shaw denied they are already champions-elect.
He said: "No, not yet
It's a long season so we'll have to see what happens.
"It's a massive test
It's never easy
We're going to work hard this week and will be ready for the challenge on Sunday."
Source : PA
Source: PA