Young earth creationism looks to the Bible as its authority for the origins and development of life, plus historical accounts of what has happened (both to human cultures and the earth) since time began. Naturalistic evolution looks to the fossil record as its authority. The fossil record is the name given to the sum total of all fossils that have been discovered. Interpretation of the age of fossils is based on radiocarbon dating, a technique that can determine the age of an organic object by studying the state of decay of radiocarbon, which is a radioactive isotope of the element carbon.
How does young earth creationism view the fossil record?
Scientists have determined, using radiocarbon dating, the age of fossils in the fossil record. Because radiocarbon has a half-life of 5,730 years—that is, it takes 5,730 years for half a sample of radiocarbon to decay—it is ostensibly possible to accurately determine the age of a thing by looking at the amount of radiocarbon it contains. The oldest samples this method can reliably measure are samples that are 40,000—50,000 years old, or less. This is interesting, considering that radiometric dating (a similar technique used to calculate the age of inorganic matter, using uranium isotopes) can account for billions of years. Both radiocarbon dating and radiometric dating have problems: they require assumptions to be made about the sample (Exactly how old is it? How much of this element was present at the beginning? How much of the element has been added / taken away over the years?) that are hard to measure. Recent evidence has shown radiometric dating to be potentially flawed, due to the ample presence of helium isotopes in certain crystals which were thought to be millions of years old, but in fact cannot be more than 14,000 years old because if they were any older the helium would have dissipated. More on radiocarbon dating can be found here: http://www.gotquestions.org/carbon-dating.html. More on radiometric dating can be found here: https://www.compellingtruth.org/radiometric-dating.html.
All this to say, even the most solid scientific methods for determining the age of organic and inorganic matter have been proven to work up to 40,000 years. The discrepancy between 40,000 years and 6,000 years (the age of the earth commonly upheld by young-earthers) is considerably less daunting than 6,000 years versus a sold "billions."
It is important to view this issue in the proper light. This is not only hard empirical scientific data we're dealing with. There is a scientific establishment. There is human interpretation of data. There is motivation to prove certain theories—both for the sake of ambition and the sake of strong emotions against religion. Human error is a thing to consider. For example, consider the coelacanth, a fish fossil that was thought to be 70 million years old. Extrapolating from the interpretation of this fossil, scientists believed that they had found an ancestor of humanity—the first fish who walked up onto the land and breathed air. Unfortunately, a fisherman off the coast of Madagascar caught a coelacanth in 1938. What happened there? Enthusiasm for the theory of evolution happened.
This is not to say that everything scientists do and say is wrong, or the data is being deliberately fudged. It is only to point out that although science presents its data as fact, it is only theory, and these theories are subject to change as new data is found (as in the case of the helium isotopes in the crystals). And this is just how it should be. New discoveries should be evaluated and data added to the make the big picture clearer and clearer. However, to say that the fossil record disproves the view of a young earth is simply false. It doesn't. It can't. If the fossil record were to stand up in court and challenge the Bible, it would fail to disprove the Bible because it simply doesn't have the hard evidence.
The Bible has given us an account of creation, and an account of the flood, that is a very good explanation for how things came to be, and how marine fossils wound up inland, and on mountains, upon every continent (Genesis 7:21–23). Discrepancies between 6,000 years and 40,000 years may seem daunting, but the fact is, God has chosen not to reveal everything in His Word, and the answers scientists have are theoretical in nature (that is, extrapolated from the data) or based on testing methods that have flaws.
It seems that either way, one must have faith: either in the Word of God, or in the theories of science.
Is punctuated equilibrium compatible with the Bible? What is the theory of punctuated equilibrium?
What is radiometric dating? Does it fit with the view of a young earth?
Questions about Noah's ark: How long did it take Noah to build the ark? How long was Noah on the ark?
Was the Noahic flood global?
Is creationism scientific?
Truth about Creation