The Canaries enter Saturday's match at Southampton needing four wins in the remaining nine games to reach the 40-point mark many managers target for Premier League safety.
Scotland midfielder Snodgrass believes Norwich would be better off than four points clear of the relegation zone had they taken their chances in matches such as last week's with Stoke - who held on to draw despite being reduced to 10 men - and called for his team-mates to determine their own fate.
"We've not been able to capitalise on the opportunities we've had," Snodgrass said.
"The last two months we've had opportunities to stay unbeaten, but we've just shot ourselves in the foot and caused our own problems.
"That's why the lads are pulling their hair out, thinking 'what's the next step?'
"We were defending well against West Ham for 86 minutes, conceded two goals within five.
"We were in front against Cardiff. We conceded two goals within two minutes. I can go on.
"We've made those mistakes, but there's no better way to make a name for yourself by scoring the winner at St Mary's or keeping a cleansheet.
"I think that's what it takes. We need leaders, we need me, we need heroes.
"We do need people to stand up and those match-winners, myself included. Against Spurs it went for myself. We need more people to want to be the match-winner and the heroes."
Chris Hughton's side have struggled away from home in particular, with just eight points and nine goals in 14 fixtures ahead of the trip to Southampton, the most recent of two wins coming at West Brom in December.
"We're doing things away from home that we don't do at home," Snodgrass added.
"We don't concede two goals in four minutes, we don't concede four goals in 15 minutes. It doesn't happen at home.
"You need to regroup as a team, because it looks like we just capitulate at times.
"The boys recognise that themselves and there's been some words changed after the game.
"The boys are passionate, they care and this is what it will take to try to get results, especially away from home.
"We all know what's at hand if we want to try to stay in this Premier League. It's up to us to do it.
"Everybody's pulling in the right direction. We're all gunning for it. We just need back-to-back results to go our way to try to kick on."
With their last four fixtures against Liverpool, Manchester United, Chelsea and Arsenal, it looks like Norwich may have to ensure their safety in the next five games - against Saints, Sunderland, Swansea, West Brom and Fulham.
Snodgrass is not yet thinking about the rest of the run-in.
He added: "I don't look too far ahead. It's hard enough trying to pick a result up on Saturday against a good side.
"We'll try to beat the opposition that's in front of us. That's all we can do.
"We need to be at the top of our game and everybody giving everything they've got to try to get a result down there.
"They're a great outfit and they've been getting all the plaudits this year and rightly so.
"They've got great talents in the team. It will be a tough job and we need everybody to be firing and to stay mentally strong throughout the game.
"Give them chances and they'll punish you."