To start, it's an Alternate History where the Axis Powers, including the Nazis, won World War II and have taken control of the United States. Imagine the fascist rule and crimes against humanity carried out by Fascist Italy, Imperial Japan and Nazi Germany, but continuing beyond the 1940's and taking place on a far broader scale in territory that was once thought to be one of the most well-defended countries in the world. There is a strip of neutral territory in the Rocky Mountains, which itself probably isn't much safer.
Many things the Nazis are doing, such as the massive genocide of blacks in Africa.
Again from the pilot, while Joe Blake is driving to Canon City, he is stopped by a policeman. As they are waiting, ash starts landing on the road. The policeman comments that it's Tuesday and that's when the hospital burns the disabled and the terminally ill... like it's something completely casual.
When Frink's sister and her children are placed in the waiting room, which has a three vents in the ceiling, a thick steel door and plastic covers over the furniture. It's a gas chamber and they end up being killed in it. That is at least done off screen. The way Laura keeps staring at those vents (one vent would be innocuous enough, but the room is way too small for three of them, and she knows it) indicates that she has a pretty good idea what's going on, but doesn't want to scare her children.
The Marshal, who kills the book store owner in Canon City and leaves his body hanging from a lamppost to be eaten by carrion birds.
The show's portrayal of Reinhard Heydrich as a scheming sociopath with no ultimate loyalty to anyone but himself and zero qualms to engaging in genocide for his own advancement is chillingly realistic. Yes, the real Heydrich really was that much of a psycho.
Juliana finds out where her sister was executed by the Japanese, which turns out to be a mass grave in the woods outside San Francisco. The camera then pans out to an aerial shot, showing that she's standing in the middle of around a dozen similar kill sites.
The premiere has Hitler seeing a film of a world where the Axis alliance collapses into nuclear war. The sight of a totally ruined world is so jarring that Adolf Hitler is rocked hard by them.
In the last episode of season 2, we see Washington DC's last moments as it gets atomized by a nuclear bomb. Smith and Helen, then loyal to the USA before their FaceHeel Turn into Les Collaborateurs by the time of the series, are horrified upon seeing the U.S. Capitol destroyed. On a personal level for the Smiths, it also doubles as a symbol of their coerced Start of Darkness that would lead them and their family to have no other choice, but to support the Axis Powers.
In season 3, the Nazis are shown as enthusiastic proponents of Tested on Humans: a black woman is forced to undergo experiments at the hands of curious Mad Doctors like Joseph Mengele to alter their skin pigmentation, and elite soldiers of the Lebensborn Super Breeding Program are seen hunting down "subhumans" as part of their training.
John Smith starts to have more frequent nightmares as he undergoes a Villainous Breakdown after the death of his son Thomas by forced euthanasia at the end of the previous season. In one he's following his son through the mist, before running into a group of Nazi soldiers led by Rudolf Wegener. The stormtroopers are gunning down civilians in Cleveland, and one of them resorts to bashing a little girl's head against a wall when his pistol malfunctions. In another one he's fishing beside a lake with Thomas, before his son evaporates and the lake turns out to be filled with rotting, bloated human corpses.
The result of Mengele's failed alternate dimension experiment in the final episode, featuring Ludicrous Gibs. At least one of the victims is a child.
The year long effects of Jahr Null hits Helen Smith with her two daughters hard after their "holiday" in the Neutral Zone. Imagine a society that encourages the youth to report any type of subversive activity to the officials, even their own families. Jenifer witnesses one such incident when a student sharply criticises their syllabus as being "not in the interest of Nazi ideology". The teacher is clearly taken aback and too shaken to questioning the status quo and quickly abides.
The Nazis have finally stabilised their dimension-hopping device after repeated failures in Season 3. So what have they been up to for the past year with that tremendous technology? They have already sent covert agents throughout the multiverse, not only stealing an abundance of technological information from other worlds, but also outright sabotaging the progress of other universes, with their victims completely unaware of extra-dimensional assassins killing them without repercussions or even warning.
After being approached by the resistance to betray her husband, Helen Smith searches John's office for answers, finding a document with detailed plans for "Phase V", the industrial genocide in the Pacific States after their looming conquest by the Reich, with stats and figures for all the numbers of "undesirables," dozens of proposed death camps sites, and schematics for the train cars and gas chambers.
Smith and Wilhelm Goertzmann's coup splits the Third Reich in two. Whilst Smith is killed and there are signs of the Reichs rule collapsing in North America. There's nothing stopping Goertzmann from either reducing North America to ash or stepping in and ruthlessly crushing resistance returning it to his control.