The Kryptonian Hegemony sprang up almost overnight approximately 50,000 BCE. This was due to the Kryptonian discovery of spaceflight and an inadvertent exposure to certain wavelengths of cosmic radiation unbuffered by their atmosphere.
Kryptonians had discovered they were virtually invulnerable. Little could kill them, or meaningfully hurt them. They turned from an exploratory species to an invasion force overnight. In the space of fifty years, Kryptonians had conquered a dozen neighboring sectors and were engaged in full-scale war with Oa, effortlessly crushing legions of their Manhunter robots. A single Kryptonian was an army. Ten of them in one sector was a war crime. Needing little materiel to begin with, the Kryptonians soon found they could fly across the stars without even needing spaceships.
Oa, their chief rivals, attempted three gambits to stop the Kryptonians. The first was the formation of the Green Lantern Corps, more flexible and cunning than their Manhunter robots. The second was to dim Krypton's main star, turning it from a small but mighty blue star into a dull, weak red star. The third was to create an army of warriors, physical equals with Kryptonians but armed with magical weaponry. For this, they turned to the Demiurge, a proto-sapient mass of energy in deep space, and the refugee First Gods living there. Together, Oa and the First Gods pushed the Demiurge into true self-awareness, and from that evolution came the hawkish, proud warrior kingdom of Asgard.
Finally, their long-term gambit: a genetic plague, slow incubating and difficult to detect. Krypton did not know it was poisoned until fertility rates abruptly reached 0%. In the space of a decade, no new Kryptonians were born, to a species with an already low birth rate. In the space of a century, the Kryptonian population dropped by 99% due to war and attrition. They huddled on their home world of Krypton to try and find a solution. Oa seeded their orbit with irradiated rocks formed from chunks of a neighbor planet, creating a web of "Kryptonite" that prevented independent space flight.
Krypton turned to science, and the few survivors rapidly made up for centuries of their intellectual dark ages. Though they were unable to repair the damage to their main star, they built domes that mitigated the worst of the lethal Kryptonite and used cloning techniques to sustain the species with new Kryptonians. For thousands of years they limped along, frequently subverted and sabotaged by Oa. Even their attempts at technology were stymied-- their Brainiac AI was corrupted and sabotaged shortly after flying off-planet, becoming a dangerous threat to much of the universe's life.
In the end, one Kryptonian scientist made a desperate gamble. In approximately 1918 CE, Jor-El boarded the last of their hyperspace ships to try and locate the ancient cousins of the Kryptonians: the Kherans.
Malfunctioning equipment led Jor-El not to Khera, but to Earth and the nation of Wakanda, and their reserve of the heart-shaped herb. From their Celestial-strong DNA he was able to engineer a cure for the Oan sterility plague. Jor-El and his brother were able to use the gene therapies and help their wives conceive one child each. Kara Zor-El was born first, and her cousin Kal-El was born some thirteen years later.
For a brief moment it looked as if the Kryptonians might survive. A race brought low, allowed to return to life and flourish once more.
It was not to be. Krypton's time had come, and their planet's doomed star was rapidly destabilizing. The House of El worked frantically, modifying Jor-El's spaceship into a pair of emergency hyperspace pods. The cousins were placed into them, Kara admonished to care for infant Kal-El. Kor-El set the coordinates for the only safe place he could think of-- Earth. A primitive backwater near the Great Void at the center of the universe, far from the machinations of the rest of the universe. A place where the House of El was honored and respected.