Here are some reasons why Christians should believe in a young earth ("young" usually being defined as somewhere between 7-8K years old).
1. The Biblical genealogies take you backward to a fixed point in history, and that fixed point is not millions of years ago.
There is no evidence that great gaps of time exist between the names in the Biblical genealogies. In fact, the two New Testament genealogies of Christ (Matthew 1 and Luke 3) show direct successions from fathers to children. And the Luke 3 genealogy takes you all the way back from Jesus of Nazareth to Adam. Jesus' chain of ancestors cannot span backward millions of years.
2. We believe in the Biblical account of creation on the basis of authority, not on the basis of empirical observation. No one, regardless of their convictions about religion, can empirically observe, or reproduce, creation.
Moses wrote the book of Genesis, and Moses was a prophet of God. Being a prophet of God, it was impossible for Moses to err when he wrote. God controlled what Moses wrote, and God cannot err, therefore Moses could not err. There are liberal theologians who claim that human beings are not capable of writing error-free material, but they say this because they don't believe in the miraculous.
The error-free character of Moses' writing means that, if Moses wrote that there were six days of creation, and each of those days were split into binary parts (morning and evening), then that is what happened. It all rides on whether Moses was a true prophet of God.
3. The natural systems of the earth cannot function correctly, or even survive, without each other.
Nature is an intimately-interconnected system of life. Vegetation cannot survive without sunlight. According to Genesis 1:11-13, God created vegetation first. The next day, God created the sun and moon (1:14-19). God also created the oceans before He created the moon (1:6-7); yet tides don't exist without the gravitational effect of the moon. The absence of tides would impact everything else.
It would be impossible for the earth's water cycle to exist without evaporation (which is caused by the sun) and tides (which are caused by the moon). So, to say the creation days were millions of years long would mean there was no evaporation or tides for millions of years.
Similarly, it would be impossible for the vegetation to survive without sunlight for days, let alone millions of years. This is a reason why there could not be thousands or millions of years between the creation-days, because of the inter-connected nature of life.
4. Moses explicitly equated the length of the Jews' Sabbath day to the lengths of the days of creation (Exodus 20:8-11). He used the same word ("yom"), in the same context, speaking about the same thing. There are no reasons present in the context to say that Moses used the word "yom" literally in the first half of the sentence, then without explanation switched to an allegory in the second half.
5. The days of creation being millions of years in length empties the phrase "morning and evening" of meaning. It would force us into a mystical interpretation of "morning and evening", an interpretation which is not in the passage. Doing that then leads to worse troubles, as that mystical method of interpretation infects other passages of the Bible.
The measured age of rocks does not prove that the earth is millions of years old, because that idea assumes a non-miraculous origin to the earth, and also assumes the absence of miraculous divine intervention in natural history.
Genesis explicitly teaches a miraculous origin to the earth. Once you accept a miraculous origin to the universe, the need for a millions-of-years-old earth dissolves.
According to Moses, God created everything already aged. God didn't create seeds, eggs, and fetuses. He created fully-grown forests, meadows, birds, cattle, bacteria, insects, reptiles, fish, and a fully grown human couple. This implies He also created chemically fully-developed fresh and salt water-systems to support all of it, and fully grown geology, which would include molecules already breaking down.
If you accept the book of Genesis and that Moses was God's prophet. there are no solid reasons to believe in a millions-of-years-old earth.