"That trademark is to stop COMPETITORS passing off as you, not to censor little girl's unrelated artwork."
The site in question can print T-shirts from submitted artwork, and that's covered by the trademark - but while the image bears the phrase, the artwork is nothing whatsoever to do with Game of Thrones, it's a completely independent work, with the phrase added as a title afterwards. The girl is not a competitor, and is not passing off her work as relating to Game of Thrones.
Given other uses of the phrase, what's needed is for someone who has in the past used it in a commercial context - preferably predating the trademark, and where the phrase is used in a significant way and is recognisable from that use (which rules out simply being used in a line in a book or whatever) - to produce T-shirts with it in the context of their work.
Then we could all invest heavily in popcorn companies (or just buy lots of popcorn) and see what happens.