Thank you Hashem, for granting me the ability to write this article!
This article is not at all meant to prove evolution (though it basically has been proven by other sources), it is meant to show that the Torah creation "story" does not imply that animals were created instantly out of nothing, and that you can indeed accept the Torah's divinity and yet maintain that biological evolution is the method with which G-d created the animals (and likely humans)1.
Firstly, the Torah makes no mention of HOW G-d created the animals, but simply that he did create them, and thus I find it quite absurd that people cite the Torah to back up their evolution-denial. This article is based off the creation model that is described in THIS ARTICLE. Another (perhaps less plausable) argument is that (as some sages have said) the Torah's creation "story" is not at all to be interpereted literally. This is one of Natan Slifkin's "rationalist" ideas, which I do not approve of usually2 but may indeed answer some questions that arise from reading this article. Also this article assumes that the first Torah chapter is to be taken literally. According to the calculations of each day's length shown in the article linked above, the plants were created after about 12 to 14 billion years, meaning plants existed for roughly 2 billion years to 500 million (I included that last number because that is what science currently suggests [regarding land plants]), and keep in mind that the calculations in that article are only rough estimates. Or perhaps the plants mentioned here are things before the first land plants. Perhaps the Torah is refering to very simple single celled organisms that wouldn't scientifically be called plants (in fact that is the most likely explanation in my opinion). Many hypotheses I have presented, but it need not matter what that verse specifically means for this discussion. Anyway, the calculations for the appearence of life are very roughly accurate, but like I said, the calculations (of each day's length, from the article mentioned above) are not perfectly accurate. It says "Let the Earth sprout vegetation, seed yielding herbs and fruit trees producing fruit according to its kind..." This might mean that in the 3rd day the plants were first sprouting, but that they were not trees until the 4th day period (approx. 2 billion to 500 million years ago). Also it's not clear what the terms mean exactly, so perhaps the very first plants (From 500 million years ago) are actually what's mentioned here. Anyway, I would suppose that (as scientific theory says, with evidence of things blocking the view of the sun and moon [like asteroid smoke]) the moon and sun being mentioned here is refering to them finally being visible, no longer clouded by things like asteroid smoke3. I'll probably ask someone about why the sun and moon are mentioned so late, and I'll probably write an article discussing that (b'ezrat Hashem, it will happen; that would make this article part 1 of a series, which I indeed will very likely continue, and likely beyond a mere 2 parts). Basically, the Torah mentions G-d creating things, but not how he did create them.
So if biological evolution is almost 100% proven (which it is) then do not be afraid to accept it. And if you are reading this (you are) may Hashem bless you!
1: I would in fact suggest that humans were very likely created from the process of evolution as well, and that the statement from the midrash saying that Adam was created from the dust of all the corners of the earth can be explained by noting how evolution happened (abiogenesis happening basically from dust, and thus animals coming from dust, therefore explaining how Adam was formed from dust) throughout the whole world and then eventually Adam was formed from the evolution of animals that were spread out across the whole world (if that makes ANY sense).
2: Slifkin has gotten into QUITE the controversy including his books being banned, and I think a lot of what he says, though not being full heresy, is still not true. Read his articles with a grain of salt. I do, however, think his idea that the Torah's creation story is not literal is potentially quite reasonable.
3: This is the main argument that I see could be made for the Torah's opening chapter not being literal; I still doubt it though. Edit: I have written an entire article (part 2) that explains this.
P.S. It's amusing how I link a video by Professor Dave, whom I previously debunked, to help prove evolution, though do not actually believe what he says about the Torah (Chas Veshalom).