For e.g. 安心するていうか。I saw this in an interview where the person expressed relief that a certain place was available.
Is there any nuance with just saying 安心する? This is one thing I can never seem to get. Adding extra words at the end that just seems redundant. Except that, ていう seems to be added everywhere by everyone like a generic individual speech tick similar to Naruto's だってばよ。
In "real-life" informal conversations among us native speakers, we do not end sentences with だ/です/ます/します/しました/だった/でした, etc. nearly as often as Japanese-learners might expect us to or have actually been taught that we do.
(Then again, native English speakers rarely speak English the way we are taught that they do in Japan, either.)
So, how do we end sentences?
We often use sentence-ending particles and/or little phrases that just kill Japanese-learners because those words and phrases do not translate easily.
「というか / っていうか / っつーか」(← three forms of the same phrase listed in the order of formality) is one of those phrases that we attach to statements to avoid directness, over-assertiveness, etc. In short, those are softeners.
「っていうか」 has a meaning and function similar to those of "I mean" or "I mean like" in colloquial English.
Thus, 「安心するっていうか」 roughly means "I feel, I mean like relieved." I hope you are starting to understand how it would not be easy to teach this in a beginning or intermediate Japanese course. That is real Japanese, but it is simply too colloquial.
As you have rightly observed, 「というか / っていうか / っつーか」 is heard everywhere. If a native speaker did not use this attachment, he would still say:
「安心するよね。」
「安心しますね。」
「安心するわ。」
Very few would just say 「安心する。」 in real life.
The grammatically valid use of というか / ていうか is when trying to convey that the information you're imparting comes from a source other than yourself. (Additionally the か part hints that further information or some sort of qualifier is coming next) BUT in modern Japanese, younger people will use these forms as a way of emphasis and/or to sound like part of their in-group. Similar to the way young Californians (used to? still?) say "like" when describing something someone was saying or doing: "She was like, soo drunk! / Can you like, go over there, or something?" .... Or similar to the way some Canadians might end their sentences with "eh?"
Another common reason for using というか / ていうか is to give the speaker time to think of something further to say/organize their thoughts about whatever they're talking about, and to let the listener know that something else is coming, so that the speaker can maintain "control" of the conversation until they've fully expressed their thoughts.
