It's not entirely clear to me what you are asking here, but I'll try to answer as best as I can. Feel free to offer any necessary clarifications.

First of all, the Turin papyrus (presumably the Turin King List not the Turin Erotic Papyrus includes a list of historical pharaohs? Yes. That's what it's for.

Here's a hieroglyphic transcription for reference.

For historical periods, it's simply a list of Egyptian pharaohs, but as you go back further the certainty of the information starts to deteriorate. Nearer the beginning of the papyrus, it lists semi-mythical pharaohs which have no presence in the archaeological record. In column 2 line 5, you can see: "King of Upper and Lower Egypt, (Menes)" 𓇓𓆤(𓏠𓈖𓇋). That's the semi-mythical first uniter of Egypt, usually identified with Narmer, although the connection is uncertain. James Allen thinks his name means "Memphite", because as the uniter he was the first Upper Egyptian ruler to live in Memphis. Maybe. The truth is that we don't know who Menes was or whether he even existed. He's a legendary figure, more Uncle Sam than George Washington. Saying that Menes is Narmer is a bit like arguing that Uncle Sam really is George Washington. What does that even mean?

Going back further, things only get worse. Column 1 line 12 has: "King of Upper and Lower Egypt, (Sun)" 𓇓𓆤(𓂋𓂝𓇳. This is the word you know as "Re" or "Ra", i.e. the sun-god, Ra. Probably not a real person. Under that you find different gods, all ruling for hundreds and thousands of years each. At the end of line 17, 𓆳𓏏𓍢𓍢𓍢, means "300 years" for Horus. In line 18, Thoth rules for 7,722 years (𓆳𓏏𓆼𓆼𓆼𓆼𓆼𓆼𓆼𓍢𓍢𓍢𓍢𓍢𓍢𓍢𓎆𓎆𓏤𓏤). Since you now have this opportunity to decode Egyptian numbers, you can calculate the reigns of other legendary kings. Obviously these are not actual reigns, because they're all way too long. What you're seeing is the same thing that you see in other creation legends, such as Hesiod and the Bible, early people were semi-divine giants made of gold who lived for centuries. I personally don't think those sorts of things were ever even meant to be taken literally. Obviously historians cannot take this seriously as a historical document.

As far as how that relates to water erosion, I honestly don't even see the connection. Even if you claim that Egyptian civilization is 13,000 years old (you can, people did live there for that long), that still doesn't put Khafre any earlier than the Old Kingdom (Column 4 line 12), and the sphinx is clearly a portrait of him. It looks just like his other statues, and it's sitting in front of his pyramid behind his mortuary temple.

In sum, the Turin King List is a dubious source for very early history, it doesn't coincide to an earlier date for the sphinx, but it does strongly support the consensus date.

What your question is essentially asking is "who did the Egyptians come from?" This question is difficult to answer because we cannot rely on ancient sources for this period nearly as much as we can for others. Much has been lost and what hasn't been is not especially trustworthy as the Egyptians had a nasty habit of just rewriting their history to fit whatever narrative they wanted. That said we can learn some information through archaeological discoveries.

We know for a fact the area of Egypt has been occupied since the late Paleolithic era thanks to the Nazlet Khater skeleton which was found in 1980. Other discoveries include stone tools along the Nile, and the mobile structures of hunter/gatherers by Waldemar Chmielewski that are believed to be from around the same time frame as the Nazlet Khater skeleton. There is also evidence of Aterian tool-making dating back to around 40,000 BC.

A variety of cultures began to emerge as we reach the 15,000 BC mark, groups such as the Halfan culture which were herders and fishermen. The Qadan culture is another noteworthy one which hunted and practiced the cultivation of wild grasses and grains. Many other cultures popped up and then vanished and had little difference (that we know of) between each other, generally using a mix of hunting and gathering and establishing small settlements made with wooden huts. But around 4,000 BC we have the Maadi culture, which has simple pottery that shows connections to the Levant, and the use of copper tools. The Naqada culture however is the most interesting and where things really begin to change. The Naqada not only cultivated food but traded with regional groups. Finds include obsidian from Ethiopia and ceder from Lebanon.

The Naqada culture continued to flourish and around 3,200 BC we have a period known as Naqada III. This period saw city state formation in the region as well as the first usage of Egyptian hieroglyphs. It should also be noted that this is when the royal seal known as the Serekh begins to be used by political leaders. The Serekh is the falcon of Horus and was used as a symbol of the pharaohs. Though little is known warfare between the rising political powers ultimately lead to three city states during this period fighting for control of the region. The cities were:

• Thinis, the capital of the tribal Thinite Confederacy

• Naqada, which was in close proximity to gold mines and was the center of the cult of Set, the god of violence and disorder

• Nekhen, the center of the cult of Horus (Horus and Set were rivals in Egyptian mythology) and interestingly the world's oldest zoo

Naqada was the first city to fall and somehow (through warfare or agreement, no one knows) Nekhen came under the control of Thinis. With Egypt now unified the ruler of Thinis, a man named Menes, became the first pharaoh. It was debated sometime who the first ruler was, Menes or Narmer. But most modern Egyptologists now identify Narmer with the pharaoh Menes. It was through this first dynasty that much of the Egyptian culture we are familiar with first began to take shape and from which the ancient Egypt we know was born.

Sources:

Midant-Reynes, Béatrix. The Prehistory of Egypt: From the First Egyptians to the First Kings. Oxford: Blackwell Publishers.

Carl Roebuck, The World of Ancient Times. Charles Scribner's Sons Publishing: New York

"Ancient Egyptian Culture: Paleolithic Egypt" Minnesota State University https://web.archive.org/web/20100601171500/http://www.mnsu.edu/emuseum/prehistory/egypt/history/paleolithic%20egypt.htm

Nicolas-Christophe Grimal. A History of Ancient Egypt. Blackwell Publishers