Welcome to the board, Mijo. It's an unusual forum name so I hope you don't mind I Googled it looking for evidence of sincerity or evangelicalism. Are you the science tutor or was that someone using the same tag?
If people find out you are sincere (I don't see the big deal, time will usually tell) you will, I'm certain, hear apologies from them. I think everyone needs reminders that we are not psychic and can't always tell by a post if it is going to be
one of those posts. If I was concerned, I'd rather not say anything than risk attacking someone falsely. I'm on a tell everyone to be nicer crusade. Otherwise I might be

as well.
It is true there are Evangelicals who frequent the skeptic boards with similar OPs. They think they are clever asking for us to prove the one thing they think is going to make skeptics say, "wow, then evolution can't be true, Jesus save me." They don't realize it's a fantasy. The science of evolution is well beyond that position where a theory looks good but might still be incorrect. I didn't think your OP was the typical Evangelical OP of this type if it is of any help to you to know that.
Evolution is a theory that passed into the 'overwhelming evidence for it' phase at least a decade or more ago and anyone but a science purist would probably say it happened 3-4 decades ago. In fact the bulk of arguments evolution deniers put forth are based on 40 year old science. While a lot of the public is unaware of the progress of science in this field, evolution deniers add the choice of actually avoiding looking for it. They make no effort to search out the current state of the science, preferring instead to repeat what they heard from some other Evangelical. It sounded good to them. It was cleverly stated, but utterly false.
Which brings me to cbish's post.
Have you really studied this? There is ton's of info that will explain what you've been asking. It seems that you've either bought into whomever you've been talking to or you already have some deep seeded opinions already. The fossil record is probably the weakest evidence of evolution we have. It just happened to be the first. Unless you have a specific question, I suggest you do your own homework; if you really want to know!
It is true, the evidence for evolution is at your fingertips (which is another reason your OP is suspected by some as we've said).
TalkOrigins has cataloged every evolution denier and ID/Creation supporter's claims and provides the science supporting why the claims are wrong as well as just about anything else about evolution you might want to know. AnswersInGenesis is the opposite. It has every unsupportable claim and false analogy you could possibly think of. The arguments are clever, but false.
I would, however like to promote a couple of approaches to explaining why evolution is overwhelmingly supported by the evidence. I encourage us to "frame" how we present data and ideas to people because it gets past some of the misconceptions more quickly.
One of those approaches is what I just said. There is overwhelming evidence supporting the theory of evolution. Because science does not seek to "prove" theories and because the word theory is confused with hypothesis by much of the public, when we don't come out and say evolution is now proven, it confuses people.
Using the terminology, 'overwhelming evidence', we remain accurate yet make it clear we are talking theory as in theory of gravity, not theory as in, "I have a theory".
The second approach is only needed when someone is making the claim ID is a viable theory. You mentioned the usual, 'science doesn't investigate the supernatural'. That's true but non-scientists don't have a clue what we mean by it and a simple explanation is heard as the
Evangelicals promote, "science is biased against religion". So when the fairness doctrine is waved or someone is claiming there is evidence for the alternative "theory" of ID, I address the evidence they speak of rather than explain why I don't need to address it.
Regarding your specific question, which seems to be you are considering the "not enough time" fallacy:
One claim of IDers is a continuation of denial of evolution rather than support of ID. That is
the claim not enough time has passed for evolution to have produced the life forms we see today. This false claim is based on some guy's fancy calculations but flawed reasoning and uses an incorrect process as the model of how genetic change is supposed to occur. See TalkOrigins for more details.
These are the things I like to point out:
The evidence from the fossil record and Darwin's observations led us to look for slowly changing features like one sees when you make a morph from one image to another on a computer. But now that we know a lot more about genetics, we know there is more to it than simply morphing fins into limbs.
The embryo has an organized blueprint with interchangeable parts. A rabbit gene that initiates a mammal eye in a rabbit fetus can be inserted into a fruitfly egg which has the equivalent gene removed and an insect eye will grow in the right place in the fruitfly larva. This experiment has been done.
If you think about what has to happen for an animal to grow an extra finger or toe, it doesn't start out as a bump and turn into a digit after subsequent generations.
We don't have people with digit stubs getting longer with each generation. A simple mutation in the gene that directs the fetus to grow digits gets you a complete digit because there are separate genes for growing all the structures. You only need a mutation in the number of digits gene. This is one reason the time argument was so flawed. The math guy didn't understand the genetic science.
So in a nut shell, the genetic record has replaced the fossil record and science is no longer concerned with the gaps.
The other main claim of scientific evidence supporting ID is the claim one can find genetic dead-ends going backward. They call it
"irreducible complexity". This has been completely debunked and the studies by Behe who is credited with this claim have been shown to be wrong. Behe didn't recognize the precursor to the bacterial flagella because he looked for structures. It turns out the genetic precursor was there all the time. The "robustness" of DNA (meaning the flexibility) meant the precursor was not structurally similar.
I've linked the website from Ken Miller below. He has done the most thorough job addressing Behe's claims. Miller also addresses the claims about "design" which lacks true logic and instead uses false logic since one cannot really conclude complexity is or isn't designed. In fact, some argue the robustness of DNA is more likely to be "not designed" because it has so many options you wouldn't need so wouldn't put there if you were designing it. Human designed things are not robust.
Answering the Biochemical Argument from Design; by Ken Miller
Dumbski has a rebuttal page,
STILL SPINNING JUST FINE: A RESPONSE TO KEN MILLER in which he puts out more of the same false analogies and unsupported claims. Still spinning is right.
There has been a revolution in genetic science in the last decade due to a breakthrough in lab technology making the study of DNA and RNA cheaper and easier. In addition, a whole new field, Bioinformatics, merges computer science with genetic science. Computer programmers routinely work with geneticists and the results have just been incredible. Spend an hour just perusing the genome research and you will find the level of detail we are now looking at is as I said earlier completely overwhelming.
So that, Mijo, is why we know evolution is how we got here.