Star Wars has hinted that Starkiller Base was constructed by the Empire, not the First Order, and that construction began shortly after Palpatine became Emperor in Star Wars: Episode III - Revenge of the Sith. The sequel trilogy launched with the introduction of a new superweapon, Starkiller Base, which was essentially the Death Star on steroids.
Starkiller Base was an entire planet that had been converted into a mobile battle station by hollowing out its core, and it was armed with a superweapon that consumed suns as a power source. The Jedi: Fallen Order game recently confirmed that the Starkiller Base planet was Ilum, an icy and lifeless world that was unusually rich in Kyber Crystals. And now the Star Wars: The Rise of Skywalker Visual Dictionary has tied everything together quite neatly.
The Star Wars: The Rise of Skywalker Visual Dictionary contains a timeline of the entire franchise that's centered around the firing of Starkiller Base in Star Wars: The Force Awakens. This confirms that the Empire discovered the planet Ilum a year after Palpatine took the Imperial throne. The Empire immediately began Terraforming and strip-mining Ilum, converting it into Starkiller Base. It seems that Palpatine's Empire was constructing Starkiller Base at the exact same time they were building the Death Stars.
On the whole, this makes sense. The construction of the Death Star used almost all the Kyber Crystals in the entire galaxy, so it should have taken a massive amount to make Starkiller Base. The Jedi Order had preserved the secret of Ilum's existence for over a millennia, sending younglings there to find Kyber Crystals for their lightsaber cores; what's more, in spite of the fact they'd been using it for so long, The Clone Wars proved that the Jedi hadn't even begun to deplete the vast deposits of Kyber.
And yet, ironically, this does cause major continuity problems. Star Wars Resistance revealed that the first attempts to construct Starkiller Base met with failure. Resistance agents discovered three ravaged worlds, each of which had the core drilled out of it. The first had been shattered; the second was split in two; and the third was whole, but completely unstable. Though the Resistance didn't notice it, it was clear that they were flying through planetary cores that were rich in Kyber Crystals. The First Order's tests didn't end there; the system's star had completely disappeared, presumably consumed by the firing of a Starkiller Base weapon. Ironically, these three failed Starkiller Base tests now clash with the Star Wars timeline; when exactly were they conducted? Ilum was the prototype, which means it should have been the first failed test.