Surely the main reason Christ is very rarely depicted in this way is simply because there is so little source material for Jesus as a child or young man (I've omitted "teenager" simply because I'm not convinced this was even a concept in classical times).
Our primary source for the life of Jesus is the gospels, and by chapter count, 95.2% of the gospel narrative is focused entirely on times when Jesus is a grown man. The gospels of Mark and John don't even have Jesus' birth or childhood.
If we're feeling generous to Jesus' young-boyhood, we can assign the remaining 5% as such - 3.3% Jesus' birth or prior, 1.5% Jesus as a young boy. Bear in mind though, that my 1.5% includes Matthew 2 (The visit of the wise men). My understanding is that this took place when Jesus was a young boy, although culturally we tend to tell this as part of the Nativity story, and people often assume he was a baby. If this assumption were correct, then all of a sudden Jesus' boyhood only occupies 0.3% of the gospels, with Jesus birth and prior occupying 4.5%.
The only verses in the Bible which are explicitly anything to do with Jesus' boyhood are Luke 2:41-55, the story of "The Boy Jesus in the Temple". There are no other primary sources still available to us which give an actual depiction of what Jesus was like as a child. Given that most of the traditional art around Jesus is focused around some sort of material in the primary source, it seems pretty unreasonable to expect that there would be anything like the same number of paintings of Jesus as a child/young adult, than there are of Jesus as an adult/baby.
To get the statistics above, I used the following figures:
Chapters which refer to Jesus as a grown man: 85
(Matt 3 - 28, Mark 1 - 16, Luke 3 - 24, John 1 - 21)
some of these don't explicitly refer to Jesus as a grown man, they are simply after the chapters which finish referring to Jesus as a child. You could possibly have a fourth category of "chapters which don't refer to Jesus at all", but this wouldn't take anything away from the point that chapters in which Jesus is described as a child are a tiny minority.
Chapters which refer to Jesus as an infant or prior to Jesus' birth (Wise men visited in Jesus' childhood): 2.75
(Matt 1, Luke 1-2:40)
Chapters which refer to Jesus as a child (Wise men visited in Jesus' childhood): 1.25
(Matt 2, Luke 2:41-55)
Chapters which refer to Jesus as an infant or prior to Jesus' birth (Wise men visited in Jesus' infancy/standard nativity story): 3.75
(Matt 1-2, Luke 1-2:40)
Chapters which refer to Jesus as a child (standard nativity - these are the only verses which explicitly describe Jesus as a child): 0.25