There are a couple of Scriptures which allude to the answer. First, when God created Adam and Eve, He told them, "Be fruitful and multiply, and fill the earth." (Genesis 1:28) Again, He told Jacob to "be fruitful and multiply; A nation and a company of nations shall come from you, and kings shall come forth from you." (Genesis 35:11)
If God were creating each person separately, why would He give Adam and Eve, and Jacob, this responsibility to multiply?
We learn a little about how God works with humans from the way He created the animals. He used very similar words when He blessed the animals saying, "Be fruitful and multiply, and fill the waters in the seas, and let birds multiply on the earth." (Genesis 1:22) And, in Genesis 1:24, God said "Let the earth bring forth living creatures after their kind."
So, we see that God created the pair of animals and made them where they could bring forth after their kind. God did the same with humans. He specifically created Adam and Eve and gave them the ability to procreate, to fill the earth without His special intervention with each one.
One important thing to note: God allows humans to make mistakes and even to do wrong things when they procreate. One example is when the angels saw human women and chose to have children with them. Their offspring were not authorized by God, but were still created using the natural processes. Those offspring became very evil (Genesis 6:2-7). They became so evil that God destroyed them in the Flood. Only Noah was "perfect in his generations”, i.e. his lineage was traced back to Adam and was not defiled by the fallen angels. He and his family were saved from the flood and continued the human race.
We see children born today who are affected by the results of Adam's sin, which includes the whole dying process. They are born with sickness and birth defects. Children may also suffer hardships due to inappropriate human intervention. We know that God loves every single members of our wonderful human race, and He will heal all deformities, sicknesses, and pain in His Kingdom (Isaiah 35). The Bible assures us that the lessons learned from these difficulties will more than be worth the hardship endured in this life (Romans 8:18-23).