So, what is the Normal-type?
Every RPG that features elemental magic also features non--elemental magic. These can be late-game or early-game spells, status moves with no elemental association, or common moves that can be obtained by most members. The most obvious one, which serves as Pokémon's inspiration, is the Final Fantasy series, where physical attacks are by default non-elemental, and end-game and early attacks are also non-elemental. However, similarly, some of the elements associated with Light, like Holy in FFI, can also be non-elemental. Normal-type covers this as well.
Normal-type is also famous for demarcation of two types from it, Dark and Fairy, the former being an in-between type to fit in the middle of Normal and Ghost, and the latter being an in-between type created to bridge the gap between Normal and Psychic. Several elemental associations like Ground's association with Sand and Flying's association with Wind also originally originated from Normal-type to be granted to these types, in addition to Kiai/Focus-based moves which later became associated with Fighting.
The History of the Normal-type:
Normal-type was supposed to be one of the worst types. The vast majority of the Normal-types you could obtain were either early route fodder, or upgrades to early route fodder. However, Normal-type was granted several additions, like access to Hyper Beam, Body Slam, Double-Edge and Explosion, as well as some of the rarest Pokémon in the form of Porygon, Snorlax, Tauros and Chansey. However, thanks to speed based critical hit mechanics, as well as the earlier Hyper Beam mechanics, combined with a lack of a proper alternative besides Ground, have helped Normal-type to become one of the best types in the competitive, which also buffed Rock-type, another early type that was meant to suck.
This quasi-unintentional mechanic combination has not helped the early route fodder, Ditto, Dodrio, Lickitung, and Kangaskhan, though powerful, was largely outclassed by Tauros. However, Normal-type enjoyed the access to the strongest, accurate moves, though those moves were also accessible for other types as well, with vast majority of signature moves of Normal-type belonging to species that don't have the type at all, like Lovely Kiss. However, Normal-type's vast repertoire of moves served as the basis for other types, which as time went on gained their counterparts to moves like Double-Edge and Quick Attack.
The Special split was something of a blessing in disguise for Normal-type. Most Normal-type were either unaffected by the split, or buffed by the split, save for species like Tauros and Chansey, whose large Special movepools were now rendered irrelevant. However, since most buffs were on the Special Defense side, this split has turned Normal-type into one of the most significant Specially Defensive types in the game. This idea was later given to Fairy, however.
Another blessing in disguise was Steel-type. Though Steel-type was designed to also wall Normal-types, their introduction have helped the metagame to shift in favour of a defensive style. With lack of a Ghost or Fighting-type to wear them down, defensive Normal-types became the significant representatives of the type, which caused the new cross-generational evolutions to shift in favour of defense as well. Tauros also gained a female counterpart, Miltank, as a showcase of the new breeding mechanics, which became an infamous wall on its own.
However, three moves, Gust, Sand-Attack and Bite left the Normal-type for sake of other types. There hasn't been much to compensate for that, but Belly Drum was accessible for some Normal-types, and the happiness mechanic introduced in this generation brought in two strongest reliable moves of the type, Frustration and Return. Hyper Beam and trapping moves were also nerfed this generation, which made Tauros useless.
The third generation was the big shift for Normal-type. Few interesting moves like Covet, Crush Claw, Fake Out, Facade, Stockpile-Swallow-Spit Up and Secret Power&Nature Power were introduced. Another new gimmicky Pokémon to reintroduce the weather mechanics, Castform was brought in. However, the biggest introduction was oddly enough Whismur, a new sound-based species with moves like Uproar and Hyper Voice. While Normal-type had other sound-based moves that were largely used for status like Growl and Screech, this was the second Sound-based species Normal-type had, after Jigglypuff-line. This brought in Normal-type the association with sound moves.
Introduction of Abilities only made existing Normal-types even more gimmicky. Forecast was the poster child of this, the earliest Ability to shift forms, but others like Color Change, Normalize and Truant were also introduced, almost all of which were terrible. Other Abilities given to this type like Run Away and Pickup were also completely useless competitively, and Cute Charm would not survive the introduction of Fairy-type.
Special-Physical split also put beam based and sound based moves firmly on the Special side, with rest of the moves being largely Physical. There wasn't much else for Normal-type however besides Giga Impact, Last Resort and Rock Climb. In terms of Abilities, Normal-types gained several new Abilities, with Scrappy being the most significant one, though it would later be distributed to Fighting-types.
The fifth generation was a mixed bag for the type. Though the type gained some powerful moves, Abilities and species, these were all overshadowed by the buffs Fighting-type got, which made former OU staples like Snorlax no longer useful, leaving OU for Chansey, Blissey and Staraptor with the occasional Meloetta. However, this generation gave Normal-type its second elemental type combination in the form of Deerling.
Another major loss for the type was the Explosion nerf. Explosion was originally given to effectively work as a critical hit to balance the weak species it was given to, but in this generation it was turned into a standard move, which made it useless.
The sixth generation introduced the big drift. Withe the introduction of Fairy-type several Normal-types migrated from Normal-type to Fairy-type. While this generation Normal-type gained Boomburst, a Special nuke move, everything else it gained were either useless, event-exclusive or species-exclusive. To add further insult to injury, the introduction of -ate abilities made Normal-type coverage irrelevant. However the Normal-type gained some janitorial cleaning like Protean to Kecleon, stat buffs to early route third stage species etc. In addition, it gained some powerful Megas like Mega Pidgeot and Mega Kangaskhan, and in this generation it finally became a secondary-type to be utilized by Elemental types.
The seventh generation was a hit and miss. With lack of new Abilities to abuse and new moves to use, it remained dry. Though it was given some powerful species, they were largely outclassed, and most of them had either more relevant secondary types, or in Silvally's case could change their types. To add further insult to injury in parallel to the Fairy drift several early Normal-types gained or became Dark type in their regional forms. With the resurgence of Steel, its old ally turned enemy, it seems Normal-type's days are over.
The Overview of Normal-types:
Normal-type is kill.
More seriously, though, with the introduction of Dark-type, its status as a Physical coverage was made irrelevant, and with the introduction of Fairy-type, its status as the primary Special wall was rendered useless. While Normal-type still gets success from its status as the jack of all trades, master of none, which gives higher HP than most types, and gives it access to wide variety of status moves, it is simply not good enough of a type.
There are few ways Normal-type can continue to be relevant. Extreme Speed is still a viable option that is not available for most Normal-types. Continuing as a secondary type is also an option, due to wide variety of moves Normal-type brings in, and the useful Ghost-type immunity it brings. Fighting-type weakness hurts, but it only hits seriously if you are a mono Normal-type... which most Normal-types are. But if you aren't, you can probably mess things up.
Normal-type is also paradoxically far too overspecialized for HP, Special Defense, and Attack, with far too little species with Special Attack focus and almost no species with Defense focus. More species with Defense and Special Attack would help Normal-type to grow further, though for all intents and purposes Normal-type is pretty complete, which is why its decline is quite terrifying.