Japanese people also have pauses in their sentences, just not necessarily as many and as long as those a learner would have. The key is not in avoiding pauses entirely to speed through sentences, it is in finding where it is natural to pause.
Instead of trying to read an entire sentence in one breath, keep your pauses but have them fall after each particle instead of randomly in the middle of words. Practice reading that way, then slowly work your way up to reading the entire sentence fluidly. This will help with your intonation and having stress in the correct places when you get to reading fluidly, too.
So, for the examples you’ve given it would go:
Btw, 三かい is read as さんかい, not みかい or みっかい.
When you’re reading a new sentence out loud for the first time, take a moment to identify where that first pause will be before reading the first part out loud. Once you’ve read that part out loud, pause and do the same for the next part, then the next, then the next, and so on until you get to the end of the sentence.
Once you’ve started combining sentence parts together, the first ones you’ll want to combine are those connected by a の, as these generally make up two parts of a whole and aren’t usually separated by native speakers (there are exceptions!)
The particle most paused after by native speakers is probably は, so that’s the best place to have pauses.
Disclaimer: Everything mentioned in this post is based off my personal experiences and observations and no official source, so if anyone disagrees with anything I say please point it out and let’s have a discussion.
Also, as many have said, you should listen to native speakers as that will help you identify the natural flow and inflections of a sentence and where pauses should fall.
TL;DR Put pauses after each particle in the sentence at first, then work your way up to reading the entire sentence as a whole.