Chris Hughton's Canaries haven't picked up three points on their travels since beating West Brom 2-0 at The Hawthorns back in December.
That was only their second away win of the campaign, having lost 11 of their 15 games so far.
That record is in stark contract to their form on home soil, where they have won six and drawn six of 16 games, losing just four times.
Norwich's latest success in front of their home fans, a 2-0 victory over fellow strugglers Sunderland, moved them up to 14th place and seven points clear of the Black Cats, who are 18th but have two games in hand.
Next up is a trip to Wales to face Swansea City, who have lost four of their last six games in all competitions but gained a morale-boosting 2-2 draw with Arsenal in midweek.
Snodgrass wants his team-mates to play without fear at the Liberty Stadium and has demanded a stronger start after shipping four goals each time on their last two visits to Southampton and Aston Villa.
The Scotland midfielder told the Pink 'Un: "We've got to take our home form into an away game - that's the only way you can build on it.
"Many managers want their team away from home to play like they do at home. That's what Liverpool have done. That's why they are title contenders. If there's any proof in the pudding, it's Liverpool.
"You need no fear and that winning mentality going into a game, and big-game players to pop up with the goals."
Snodgrass admits it is ultimately up to the players to deliver the goods, adding: "You can sit here and speak for as long as you want, the manager can put as much tactics up as he wants, but it's you as a footballer, taking it into the game.
"We did it at Swansea last season. We've just beaten Sunderland. Confidence should be high. But it took us until the 85th minute at Southampton to have a go - and that's not good enough, especially for the fans that travelled down.
"We'll work as hard as we did last week on the training ground this week, and the manager will know we're only a performance away from getting a good result on the road.
"It's about the leaders on the park and knowing the situation. Five minutes before half-time, five minutes after - those are the key moments that change games sometimes, like at Cardiff.
"When we're conceding goals we are giving the other team confidence and it takes the stuffing out of us. So if we can stay in games until late on or lead, then I believe we will get the result."