Rhinos, who won the Grand Final only two years ago, have 10 games left to save their top-flight status.
Friday’s 36-10 defeat at leaders St Helens left them second from bottom in the table, two points behind London Broncos and ahead of Hull KR only on for and against.
Leeds face a crucial game at home to Catalans Dragons on Sunday and Dwyer believes there is more pressure on them than other teams in the relegation dogfight.
Fifth-placed Salford Red Devils are only three wins clear of Hull KR and Rhinos and Dwyer pointed out: “There’s six teams around us that are all in the same situation.
“I just think it’s the magnitude of our club, that everyone thinks ‘Leeds Rhinos are going to get relegated’, so there’s a bit more pressure on us.
“When we get a win no-one really looks, but when London and Hull KR are winning everyone starts jumping around.
“We are not naive to the situation we are in, but the fact of it is it’s all in our hands.”
Though a win on Sunday would be a boost to Leeds survival hopes, Dwyer insisted they have to focus on performance.
“We have got to take it week by week,” he added.
“Two weeks ago when we beat London and Wakey and we were in with a chance of beating Wigan we thought we were in a good spot.
“It has all changed within a week. London have had a good win, Hull KR have and that’s just the way it is.
“Fortunately, it’s in our own hands. We’ve just got to focus on ourselves and we have got to deliver.”
He added: “All the way through this season, we have competed with most teams, but if anything we have been our own worst enemies.
“We have got ourselves in a situation which isn’t great, but if you put yourselves in a situation it means you can get yourselves out of it.
“That’s what we have got to do now. It is crunch time and we have got to start delivering as individuals.”