Lambert has come in for criticism after Villa slumped to three consecutive defeats following last week's loss to relegation candidates Fulham.
Villa now sit 13th in the standings, just seven points clear of third-bottom Fulham, and could end the weekend just four points above the drop zone.
Speaking ahead of Villa's must-win clash at Crystal Palace, Lambert said supporters needed to be realistic about the club's plight.
"The history and expectancy is there, but there has to be that realism that this is where it is at the minute," said Lambert.
"There's a lot of stuff people don't see. Am I happy where we are? No. Did I come to Aston Villa to do this? No.
"This club should be higher, there is no two ways about it. It has great tradition and great pedigree.
"I knew the first few years would be really hard because of what we had to do. I knew that, but we have to get this club up there.
"I'm certainly not sitting here thinking it's good. We have to do everything we can to get this club up the ladder. It has to be."
Villa's run home includes games against Southampton, Swansea City, Hull City, Manchester City and Tottenham.
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
