Well, let us get the facts straight from the scriptures: -
Exodus 1:1- 13: - Israel's Suffering in Egypt
1:1 Now these are the names of the children of Israel who came to Egypt; each man and his household came with Jacob: 2 Reuben, Simeon, Levi, and Judah; 3 Issachar, Zebulun, and Benjamin; 4 Dan, Naphtali, Gad, and Asher. 5 All those who were descendants of Jacob were seventy persons (for Joseph was in Egypt already). 6 And Joseph died,
[1] all his brothers,
[2] and all that generation/age.
[3] 7 But the children of Israel were fruitful and increased abundantly, multiplied, and grew exceedingly mighty; and the land was filled with them.
8 Now there arose a new king over Egypt, who did not know Joseph. 9 And he said to his people, "Look, the people of the children of Israel are more and mightier than we; 10 come, let us deal shrewdly with them, lest they multiply, and it happen, in the event of war, that they also join our enemies and fight against us, and so go up out of the land." 11 Therefore they set taskmasters over them to afflict them with their burdens. And they built for Pharaoh supply cities, Pithom and Raamses. 12 But the more they afflicted them, the more they multiplied and grew. And they were in dread of the children of Israel. 13 So the Egyptians made the children of Israel serve with rigor. 14 And they made their lives bitter with hard bondage — in mortar, in brick, and in all manner of service in the field. All their service in which they made them serve was with rigor.
How many years passed before the king who did not know of Joseph became Pharoh is not given to us. By my calculations the last of Joseph's brothers died around 63 years after Jacob entered Egypt. We know that Mose was around 80 years of age when he led the Israelites our of Egypt. This leaves around 287 years between these two events for the oppression of the Israelites to begin and to finally increase where the Israelites' complaint against the Egyptian oppression reached God and God called Moses to lead the Israelites out of Egypt.
Remember that there were four descendant generations that went down to Egypt with Jacob. Some of the fourth generation were still very young children when they entered Egypt.
It is not always wise for us to assume that the conditions that the Israelites found themselves in being oppressed by the Egyptians happened quickly after Joseph died.
As the Israelites lived as a separate nation within the boundaries of the Nation of Egypt, as they grew in number, that the Egyptians would become concerned about their numerical strength. Israel's dependency on God would have been a concern and so the plan was put in place to limit the number of male children born to the Israelite's women to reduce the perceived threat to Egypt itself.
The statement that shortly after Israel went down to Egypt is not based on any facts and is just someone's misguided opinion as to when all of the events in Egypt took place.
[1] From my calculations, Joseph died around 55 years after Jacob took the rest of his family down to Egypt.
[2] The bible is silent on when Joseph’s brothers died, but from an estimated timeline of their births, the years they died, from what can be determined from the Book of Jasher, it is probable that the brothers started dying around 12 years before Joseph and all had died over a period of around 20 years or so.
[3] The Meaning of “hasdowr” is an age, i.e., the age of Joseph’s and his brothers and their sons.
