There are hundredsa millionsa fossils in thousandsa museums worldwide, and plain evidence of many geological events that couldn'ta occurred in a short time(erosion of the Appalachian Mts, the formation of "devil's tower" of "Close Encounters Of The Third Kind" fame) all over earth. And over 9/10 of all animal species that ever existed are now extinct. But Scripture sez that at least one pair of EVERY species of animal & bird was aboard the ark, so mosta the extinctions had already occurred.
As for "neanderthal man", there's no dispute that "he" existed, nor that he built shelters & used fire. But does that mean he was a man? Not at all. No man has ever built a more-complex structure than a common beehive or weaver-bird nest. God built this ability into them, but He didn't make them intelligent as He did us. Bees "speak" thru their "dances" & birds thru their "songs", but they can't read one worda Scripture. God coulda simply added the ability to use fire to those which He built into neanderthals.
And there's the little problem of the speeda light. the farthest object we can see with the unaided eye is the Andromeda Galaxy, which is 10 times the siza the Milky Way & is so distant that it takes light some 2.5 MILLION years to reach us. God has allowed us to discover that the maximum speeda light is some 186 K miles/second in a vacuum. We can slow light down to about 38 MPH in certain transparent gels, but we cannot speed it up past the max it reaches in a vacuum.
I believe GOD has preserved all those fossils & geologic evidence to show us empirical. tangible evidence of past ages or epochs. Simple fact is, were it not for GOD'S WILL, there'd be no fossils, etc.
Do I believe in evolution ? NO. If evolution were true, there'd be no bacteria, protozoans, etc. They woulda all evolved into "more advanced" life forms. And neanderthal & other 'cavemen' weren't made in God's own image any more than gorillas are. Only HOMO SAPIENS fills that bill.
Fact is, the earth & the universe are QUITE old. No one can prove otherwise.