The "proof" of the earth's age given by "science" is based upon radio carbon dating...which is already demonstrated to be highly unreliable and worthless in the context of true scientific approach. Lets put it this way, when you can carbon date an animal/object that you know is a year or two old and the test results show it to be 25,000-30,000 years old etc you are using a method that is worthless in terms of proof of anything...except proof that the carbon dating is worthless as a measurement.
Then why do "scientists" keep using it? Simply put, most dont wish to even consider the alternative origin/age of the earth and they havent been able to invent anything else that fits the theorys they cling to and that all such "science" in built upon.

Also, to suggest that dinasours are proof of millions/billions of years of earth existence is rather flawed by the fact that dinasour and human fossil evidence DOES in fact exist in the same layer of strata. These discoverys are often disputed by those who cling to other hopes and dreams. The same "science" that clings to carbon dating maintains this can't be true. But that should surprise no one. With this type of "science" the facts/results matter little when they point in any direction other then the one already chosen.

The real issue has little to do with the actual age of the earth. The earth's actual age really matters not EXCEPT for one thing, if there is a Creator God and His Word is accurate then other factors must come into play....judgement, accountability, etc. Thats where the problem is for mankind. We as humans would prefer not to be accountable to anything outside of ourselves. If we can "reason away" through "science" an origin and subsequent 6-7000 year history as outlined in Scripture then we can also reason away our accountbilty to specific laws/principles and inevitible judgement by a greater authority and any potential eternal outcome.