The city's proposal was yesterday delivered to the International Olympic Committee's headquarters in Lausanne, Switzerland.
It also emerged that Glasgow's bid had been given a boost with the withdrawal of a major rival, the Polish city Poznan.
According to the YouGov poll, 62% of UK people supported Glasgow's 2018 Youth Olympic Games (YOG) bid, with that figure rising to 68% in Scotland.
The same number of Scots also believe staging the YOG would continue to inspire young people to become more involved in sport and physical activity.
Glasgow's 2018 team said it was seeking to capitalise on the opportunities created by the London 2012 Olympic and Paralympic Games and the Glasgow 2014 Commonwealth Games, as well as the city's ongoing international sporting events programme.
The bid has already won the support of Prime Minister David Cameron and Olympians including Dame Kelly Holmes, Sir Chris Hoy, gold medallist rower Katherine Grainger and Glasgow's silver medallist swimmer Michael Jamieson.
Lee McConnell, also a European and World Championship medalist, said: "To be bidding for the Youth Olympic Games after the wonderful scenes everybody enjoyed at London 2012 is a real opportunity for the UK to continue its Olympic journey."
The IOC will announce its short-list in February, with the winning bid declared in July 2013. Glasgow's rivals include Buenos Aires in Argentina, Guadalajara in Mexico, Medellin in Columbia and Rotterdam in the Netherlands.
Poznan, which withdrew on financial grounds, was narrowly defeated in the race to host the 2014 Games.
City council leader Gordon Matheson said: "Our submission to host the 2018 Youth Olympic Games is a compelling one, and it clearly has the support of people across the UK. This bid has young people at its very heart. It demonstrates the very best Glasgow has to offer on both a sporting and cultural level and reflects our reputation as one of the world's leading sporting and cultural cities."
Humza Yousaf, Scottish Government minister for external affairs and international development, said: "Scotland has a growing and deserved reputation for staging successful world class events and winning the right to host the Youth Olympic Games would be a fantastic opportunity to showcase Scotland on the world stage."