Trying to work the specifics out, but Palpatine and his rise in power are much more referred to and some hinted at at than making up a large portion of screen time. Its through some casual conversation, allusions, and often his scenes with Anakin that we get a real idea for what happened.
Palpatine was a wealthy trade businessman/philanthropist who more or less was an initial non-entity in the Senate because he had no real allies or enemies. He got elected primarily because he had no contact with any of the squabbling factions in the Senate, and thus promised to be a Chancellor (President) free of the usual politics and bureaucracy, which had become a major problem with various former holders of the office resigning or serving only one term due to scandal or not being powerful enough against the factions.
When the war is going badly, he nationalizes a select few of his former businesses (which include ships, weapons, and mercenaries, which earns some grumbling in the senate about conflicting interests and supposed corruption, but the citizens in the Republic are so desperate for some ground they fully agree to it. The nationalized businesses end up being crucial in the second film as part of a distraction of the Mandalorian coalition forces, while a select group of Jedi and Republic forces destroy their biggest clone facility.
By the third film, Palpatine is openly beloved by many citizens for helping to save the Republic from invasion, with the cynical members of the senate figuring that he'll likely re-privatize his old companies with interest money from the war, and retire from public office even richer. Except pretty much no one realized the full scope of Palpatine's plans. It turns out that he had much more ownership of the nationalized companies than realized, and his continuing monetary influence spread so far wide across the armed forces during the war, that the vast majority are loyal largely to him and his agents rather than to the Senate.
Palpatine makes a public address to the Republic that due to the fragility of the peace, the tragic loss of the Jedi due a Mandalorian (actually Palpatine aligned forces) sneak attack, and to avoid any potentially dangerous governmental lethargy during another possible invasion, he will remain Chancellor (President) for life. Across the Republic, this news plays to cheering crowds. In the crowded Senate hall, there is only dead silence as the full weight of his words sets in.
In the second to last scene of the film (with Lars/Kenobi with Luke on Tatooine and Bail/Darelda on Alderaan with Leia following) Palpatine makes his formal address to the Senate, announcing that while he will remain in his position for life, he makes assurances that Republic will still thrive, the Senate will continue, and that the remaining criminals of the war (seemingly Mandalorians) will not go unpunished. While this speech goes on the audience sees various scenes: the creation of vast amounts of stormtrooper uniforms, various Jedi on scattered planets being hunted down, Anakin's placement in a suit, and a green laser being tested on a small asteroid with great success. Palpatine ends his speech with:
The rewards we have earned at this moment can not and will not be forgotten. Members of the senate, citizens across the galaxy, and soldiers of this great Republic, it is the ingenuity and sacrifice of so many that allows us to stand in a glorious era. It is a period of new peace, of new prosperity, and of new order... for all time.