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:
Very few would just say 「安心する。」 in real life.