I am not sure what model you have or if Epiphone even makes a child's guitar, but from my limited experience they seem to be what I would call a midway point at the nut. The nut is the plastic piece high up on the neck that the strings feed through. Different guitars, different nut width and therefor neck width and string spacing.

If it helps, I started the same way and this seems to be common. In addition, I found from my experience, guitars we had at the house which had very narrow necks and therefor difficult for me to play without string buzz (actually everything buzzed at the beginning) are now quite playable. I think for most of us, it just takes time to get our fingers used to proper positioning.

So I wouldnt give up on the Epiphone just yet.