One of the ways of reconciling the apparent age of the universe with the traditional religious chronology is by saying that God created the world looking older than it actually is. This was first suggested by Philip Gosse in his 1857 book Omphalos: An Attempt to Untie the Geological Knot. The Omphalos Hypothesis has remained controversial because it is unverifiable and runs counter to many people's idea of a benevolent God. It does, however, solve all of the problems of an ancient looking world and universe.
Personally, I part ways with R. Nosson Slifkin over this and have no problem with the Gosse Theory. Does this contradict the way God is supposed to work? I don't know. My general response is simply "Could be." Can the world have been created looking old? Could be. Would God make this elaborate fake-out to fool us into thinking the world is older than it really is? Could be. Hey, I don't claim to know how God works. Maybe this somehow fits into his plan. How am I supposed to know? Granted, plenty of classical Jewish scholars have rejected theological theories based on other theories of how God runs the world. But I'm not convinced that there is a solid argument either way on this. Others are free to disagree.
Will most people find this hypothesis convincing? If they already believe in the Torah, as I do, then they might also say "Could be." If they are of a more skeptical nature, whether they believe in the Torah or not, then I doubt that they will see this as a particularly plausible possibility. It does, indeed, require a stretch of one's imagination.
But in my book, this theory is not only unverifiable but also unfalsifiable. So it could be. I believe in plenty of things that are not empirically provable or disprovable.
Some have argued that since this was first suggested by a Christian, therefore Jews may not accept it. That's just wrong on so many levels that I won't bother responding.