Why does Allaah the Almighty punish a Christian child who dies without embracing Islam? Will you please clarify this point for me, because it worries me, and makes me doubtful and at a loss?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

Please note that Allaah The Almighty does not punish a person just for having been born to Christian parents. Therefore, any child who is born to non-Muslim parents and dies before reaching puberty will be from the people of Paradise. This is according to the soundest opinion of scholars, for Allaah The Almighty Says (what means): {And never would We punish until We sent a messenger. } [Quran 17:15]

Also, Allaah The Almighty Says (what means): {and no bearer of burdens will bear the burden of another.} [Quran 6:164] 

Samurah narrated  that the Prophet, sallallaahu ‘alayhi wa sallam, said that he had a vision in which he saw the dead children of the believers and the disbelievers in a green garden, surrounding Ibraheem (Abraham) . [Al-Bukhari]

Based on the above, along with other evidence, An-Nawawi said that the children of the disbelievers who die before reaching puberty will go to  Paradise.

Accordingly, you should now realize the invalidity of the premise that has caused you to have obsessions – as you say. Praise be to Allaah The Almighty, there is nothing confusing about this issue. A man is punished for continuing to be a disbeliever after having reached the age of puberty and after being able to contemplate and seek knowledge and evidence. Furthermore, an adult would not be punished until he has been educated beyond excuse of ignorance and doubt, but still refused to accept Islam.  

Allaah Knows best.