There is evidence for the biblical Exodus and for the biblical Moses. Not for every bit of the story but for enough elements to make it plausibile.
Before we begin. The earliest reliable mention of Israel as a people is the Merneptah Stele, a rock in Egypt listing the achievements of an Egyptian Pharaoh Merneptah which is dated to 1205 B.C. He refers to the Jews as a people group in Canaan (modern day Israel). The word he used implies Israel is not firmly established. But they must have commanded a significant chunk of territory to get a 'shout out' from the Egyptian Pharaoh.
But moving onto the purpose of this post, I wanted to start with Moses name. The Bible says the Jews left Egypt 480 years from when Solomon began building the temple. (Historians think Solomon's father David ruled around 1000 B.C., Solomon around 970-950 B.C.). So that makes the Exodus, according to the Bible, around 1440 B.C.
Starting with the Bible's own dating of the Exodus then, what was going on in Egypt in 1440 B.C.? It was the 18th dynasty. A lot of Pharaoh's named ___moses. (Note: Ancient Greek puts a 's' after every male name. The Bible comes to us today by way of Greek for the most part. But Hebrew and Egyptian did not put 's' at the end of male names. The original names of Moses and Pharaoh's like Thutmoses would have been Mose / Moshe and Thutmose.) The text of Exodus itself says Pharaoh's daughter named him Moses because she found (drew) him out of the water--the basket in the Nile. In Egypt at this time, Mose or close variations were normal names. They meant approximately 'son'. Lots of Pharaohs had it as part of their name meaning 'son of____insert Egyptian god'. It's possible that Pharaoh's daughter gave Moses a more elaborate name meaning 'son of something' and the "something" was later dropped off, but maybe not.
Final point on Moses's name. This has less consensus but the Ancient Egyptian word for water was approximately "Mo". So it's possible that Pharaoh's daughter in Egyptian intended to name Moses "son of the water" because she found him in the water. And you can get something like Moshe (or maybe Momose ) with an Egyptian name meaning that. And the first bit was dropped because it was meaningless in Hebrew. In Hebrew the name Moshe means "Drawer out" or figuratively "the deliverer". (And Moses apparently is a very unusual name in Hebrew, not something Hebrew speaker would be likely to come up with, happy to be corrected by any Hebrew speakers.) So the name Moses seems to have been an Egyptian name but which had symbolic meaning in Hebrew.
This makes it weirder for Jews literally making up the Old Testament while they are in exile in 400 B.C. in Babylon (what the accepted theory is) to know the ancient Egyptian language of this period so well. The text in Exodus reads like the author(s) is actually unaware that this is actually an Egyptian name.
But I'll be honest, so far this this evidence is weak.
Secondly, the Amarna Letters are letters written from Canaanite kings to a succession of 3 Pharaohs (the third being King Tut). They were written between 1399 - 1300 B.C. The subject of most of these letters is a new group of people invading Canaan. The group is referred to by a few names, but Habiru seems to be the main one. This is obviously close to the word Hebrews. The tone of these letters gets more desperate in the later instances, seeming that all of these vassal kingdoms of Egypt in Canaan would be gone if Egypt did nothing.
(If you know anything about King Tutankhamun, you'll know his father Akhenaten tried to change religion of Egypt to monotheism and build a new capital, Amarna, in the middle of the desert. And it was apparently built very fast, and the entire court of the Pharaoh was moved there. Perhaps this is why he didn't send help to the Canaanites, busy with his new capital and state religion. Also interesting if you consider the monotheistic Hebrews could have left in living memory of his reign. Monotheism was very unusual in the ancient world, there was usually a god for everything, across cultures. )
But this is still weak. There isn't even a consensus among historians that the Armana Letters are talking about the Jews. But the timing is exactly right. The name is even almost right.
Thirdly, there is a tomb for a guy named Rekhmire who was a mayor of Thebes, then governor of a province in the 18th dynastry (dating to around 1400 B.C.). He decorated his tomb with scenes from life (with commentary). There is one scene of slaves, the commentary on the wall literally says they were slaves (dark-skinned slaves from Africa and lighter skinned slaves from somewhere in the Middle East) making mud bricks using straw, exactly how the book of Exodus says the Israelite slaves made their bricks.
Finally, for brevity's sake, I'll conclude quickly with two more points. There is a famous Egyptian papyrus scroll called the Ipuwer Papyrus which contains a poem called the Admonitions of Ipuwer which seems to describe the world turned upside, 'the river is blood', 'servants / slaves making off with the people's riches', begging the king or a god to restore order. Part of it is lost. It dates to the 18th dynasty or possibly earlier. This may finally be an allusion to the Biblical Exodus from an Egyptian point of view. But who knows?
And around 1500 B.C. (a century before the Bible and I argue for the Biblical Exodus), a group of Semitic foreigners from Canaan (Hyksos) had conquered and controlled Egypt for a century were finally expelled. These folks lived in the north of Egypt exactly where the Bible says the Hebrews were. And my general understanding is it's likely the Egyptians, being ticked off and making no distinction between the Semitic Hebrews and these folks, took massive numbers of slaves (refugees, prisoners of war) as they expelled these folks. That history jives well with what the Bible says about the enslavement of the Jews.
Obviously what is missing is finding the garbage of a huge group of people traveling in the desert, dating to the 15th century B.C. That hasn't been found. And the Jews may have 'wandered in the desert' in Saudi Arabia where archaeologists are forbidden to dig (ancient Egyptian forts and trade routes went over the Nile delta, southern Israel, northern Sinai area, seems strange that a people group of 100,000 of more could escape the gaze of Egyptian for 40 years in the relatively small Sinai desert).
But this is hopefully some info that lends plausibility to many elements of the Biblical story. The smoking gun is still missing (maybe because it's in Saudi Arabia).