Double Olympic gold medallist Dame Kelly Holmes joined young athletes and organisers to launch the bid at Kelvingrove museum in the city.
A range of sporting and cultural performances accompanied the bid, which is competing with Buenos Aires, Rotterdam and Poznan.
Dame Kelly said: “A Youth Olympics here would be an incredible way to kick-start the careers of the next generation of young athletes.
“They have got world-class facilities, a great history of hosting top sports events and some of the most passionate fans on the planet.
“Glasgow 2018 is in a great position to make a real contribution to the future of the Olympic movement.”
If successful, the youth event would form part of the legacy of the 2014 Commonwealth Games in Glasgow.
The bid is a partnership between the Scottish Government, Glasgow City Council and the British Olympic Association.
City council leader Gordon Matheson said: “Young people are at the very heart of our bid and will be our strongest voice as we go all out to bring the Youth Olympic Games to these shores for the first time.
“A successful campaign would continue the legacy created by the London 2012 Olympic Games and Glasgow 2014 Commonwealth Games, helping us to continue to work hard together with young people to help them build the bright future they deserve.”