I was wondering if the possible crossing could be on the eastern side of the Sinai peninsula?
Such a crossing would not match the account on numerous factors.
"Then the children of Israel journeyed from Rameses to Succoth, about six hundred thousand men on foot, besides children. A mixed multitude went up with them also, and flocks and herds - a great deal of livestock" (Exodus 12:37).
"Then it came to pass, when Pharaoh had let the people go, that God did not lead them by way of the land of the Philistines, although that was near; for God said, "Lest perhaps the people change their minds when they see war, and return to Egypt." So God led the people around by way of the wilderness of the Red Sea. And the children of Israel went up in orderly ranks out of the land of Egypt. And Moses took the bones of Joseph with him, for he had placed the children of Israel under solemn oath, saying, "God will surely visit you, and you shall carry up my bones from here with you." So they took their journey from Succoth and camped in Etham at the edge of the wilderness" (Exodus 13:17-20).
From these passage, we learn that the first leg of the journey went from Rameses to Succoth and the from Succoth to Etham which is described as being on the edge of the wilderness. We know where Rameses is at from archeology. It was located on the edge of the Nile river delta.
"Speak to the children of Israel, that they turn and camp before Pi Hahiroth, between Migdol and the sea, opposite Baal Zephon; you shall camp before it by the sea. For Pharaoh will say of the children of Israel, 'They are bewildered by the land; the wilderness has closed them in'" (Exodus 14:2-3).
After crossing the Red Sea, we read:
"So Moses brought Israel from the Red Sea; then they went out into the Wilderness of Shur. And they went three days in the wilderness and found no water. Now when they came to Marah, they could not drink the waters of Marah, for they were bitter. Therefore the name of it was called Marah" (Exodus 15:22-23).
The Wilderness of Shur is another area well documented in archeology. It is the desert area on the western edge of the Sinai peninsula, just east of Egypt's traditional borders.
"Then they came to Elim, where there were twelve wells of water and seventy palm trees; so they camped there by the waters. And they journeyed from Elim, and all the congregation of the children of Israel came to the Wilderness of Sin, which is between Elim and Sinai, on the fifteenth day of the second month after they departed from the land of Egypt" (Exodus 15:27-16:1).
The total time was for the first portion of the journey was one month. "They departed from Rameses in the first month, on the fifteenth day of the first month; on the day after the Passover the children of Israel went out with boldness in the sight of all the Egyptians" (Numbers 33:3). While it might appear slow to people of today with their cars, busses and planes, the distance is very reasonable for a group of probably over 2 million people with livestock to travel if we note that they traveled down the western leg of the Red Sea.
If someone wanted to argue that they crossed the eastern leg of the Red Sea (the Gulf of Arabeh or the Gulf of Aqaba), then they would have had to traveled the northern side of the Sinai Peninsula to leave Rameses. This route would have taken them along a major caravan route -- something the Bible specifically stated God did not let them do. Second the route would have lead them past the southern border of Canaan and out into the Arabian desert -- thus incurring the claim that God was unable to lead Israel in the right direction. Finally, read the account again and realize that Pharaoh pursued Israel in a matter of days (shortly after they buried their dead). The speed required for this large population to travel such a distance is not possible.
Typically people wish to claim that Mount Sinai was on the Arabian Pennisula. Believers in Islam would love to claim that Mount Sinai is in Arab territory. But there are subtle clues that must not be overlooked. The wood used for the tabernacle and its furnishings was Acacia wood (Exodus 25:5, 10, 13, 23, 28; 26:15, 26, 32, 37; 27:1, 6). Acacia is the only hardwood that grows on the Sinai Pennisula and it only grows in that region. Thus it makes perfect sense that God specified a local hardwood to be used for the construction of His tabernacle.