Rabbi Dovid Gottlieb addresses this issue in many shiurim (check out Evolution & the Age of the Universe for example).
Background of approach
His general approach follows the general chareidi approach (he is a Bostoner Chassid teaching in Ohr Samayach, indeed due to his background many Rabbonim will refer you to his lectures if you ask this question), and has done a lot of research into the scientific reconciliations such as Dr. Schroeder's relativity solution, and uses his background as a professor of mathematical logic with special interest in philosophy of science to deal with this in a manner that doesn't compromise Torah positions at all but is still palatable to someone who wants rational, logical solutions.
Summary of approach
To summarise what he says, the consensus among the gadolei adar is that the 6 days of creation are 24 hours (our hours) each. I've personally asked him if he holds that we pasken hashkafa and he said yes, referencing several sources in Rambam. There is room for interesting resolutions, that the history of the universe did play out, but faster, for example, but the 24 hours is important.
So we have to now deal with fossils, and the general solution offered is the same as Lubavitcher Rebbe (see Yaacov Deane's comment below for more on this citation); the universe was created aged rather than embryonic, in the same way that Adam was created aged, as that of a man of 20 (Bereshit Rabba). Indeed, it would take 13.8 billion years to make a universe like ours from an embryonic state, and dinosaurs would be an important step in the evolution of the body of man, but Hashem skipped all that (in some sense) and created year zero 5783.5 years ago, with a universe "aged" 13.8 billion years.
Dealing with a common issue with this approach
One might ask, isn't this deceptive? He makes it one way, but makes it appear another, and (as some people claim) He did so to give us free will - this makes people uncomfortable. Rabbi Gottlieb gives the philosophical answer: Hashem isn't deceiving us because He told us in the Torah how old the universe is. The reason the world looks older is irrelevant, whether it is purely for free will, or simply because Hashem loves nature (Derashot HaRan 8:63a), or any other reason.
So, the rational thinking Torah Jew doesn't have to have a position on dinosaurs. He doesn't have to say "they didn't exist and anyone who says so is an apikorus" (see this and this answer), but he also shouldn't feel boxed in to insist that they did exist otherwise he's an anti-science irrational religious zealot, even though you'll certainly find authorities holding by both sides (as always CYLOR). He can simply say it certainly looks like they existed, and the science done on their bones paints a very cool story of what they were like and one can be impressed about that in the same way all of nature impresses one to the awesome ways of Hashem*. If he wants to follow a strong opinion in the traditional Torah world that the days were 24 hours, then he has philosophical feet to stand on.
Remember, rational thinking knows limits. Science is not history and historical theories are never theorems. Torah tells us absolute truths but it also leaves out a lot of the history in its pshat. So what actually happened is something we currently don't know for sure. Rabbi Gottlieb places a lot of stress on not getting too worked up about this. As a younger man I found that hard to hear; I wanted to know! I still do, but it's clear that there's no point taking a definite position without definite facts, and that Hashem doesn't have to have done it the way I think would be the most sensible or cool. As cool as dinosaurs are**, I'm sure if I ever find out how Hashem did it, I'll agree He did it in the most fantastic way, far beyond anything I could have imagined :)
*I'll say from experience that there are many traditional households that do not hold like this, and one should be respectful. Scientism has been a damaging enemy to the religious world over the centuries and some people have felt it more than others, and have their own reasons for not wanting to teach their children about dinosaurs.
**Do note though, the world of the dinosaurs is an especially hard one to reconcile with the idea of goodness. In a world where there were no humans with free will to accomplish a great good that makes creation worth while, for a hundred and fifty million years there was violence and horrific death we can't imagine, an ever ongoing arms race of size and teeth and poison and claws and every nasty horror in nature...