[ScylaX]

Watamote and NHK are mainly about social outcasts dealing with their bad ways from which tragicomic comedy arises. And they're both kind of eerie at times so I really don't know if that's the kind of thing you'd want to watch. Watamote is relatively episodic while NHK explores the various conditions of a shut-in by the mean of the main character till reaching a denouement.
Watamote would have been a good recommandation but the main character's situation basically doesn't move an inch and each of her tentatives to "get popular" ultimately falls appart in a burst of embarrasment. And NHK's setting clearly not your casual teenager's school life, it's a series about being independent, dealing with society and living as an adult.

There are a handful of shy and timid characters you can relate to but anime casually dealing with it and revolving around that aspect as a narrative basis aren't really numerous. Usually they're just comedy.

Similarly to Kimi no Todoke, I think Kids on the Slope is not a bad recommandation. Well the shyness of the protagonist is mainly just an aspect of the plot rather than the main "engine" of the narration but it's present as an underlying dynamic for a good part of the show, and you have a quite relevant "Lone and withdrawn character unexpectedly making a friend, keeping a somehow fragile relationship and slowly opening to others" intrigue that sort of matches your request. If you're looking for an inspiring story about a timid individual blooming by the mean of a sincere friendship, that tells you quite some things about "life", with a considerable accent on relationships and such, I think that's what you should watch.

What you want is really specific so it's hard to find suitable suggestions tho