In some Asian countries where English is not native or official languages, some universities are named XXX(Name of Place) Normal University in which the majority or all of the programs are intended to educate students who are expected to work as teachers after graduation. But there seems no such institutes of the same purpose so named in English-speaking countries.
My guess is that native speakers of English in the US would just think "oh, okay, so that's the name of the university" and it wouldn't present a big problem. However, they are very unlikely to realize that "Normal" signals that it's a school for training future teachers.
You don't see (many?) schools in the US with "Normal" in their names nowadays, but many if not most famous US universities grew out of teacher-training schools and originally had "Normal" in their name. I'm in my mid-thirties and wasn't aware of this until I started hunting for academic jobs and spent a lot of time researching the universities I was applying to. To be honest, my reaction was to think that the "Normal" names sounded a little silly (as if these were being compared to "abnormal" schools), and I suspect that other people who are unfamiliar with the concept might feel the same way. (And I suspect that most people in the US fall into the "unfamiliar with it" category, or at least most people under age 40 or even under age 60.) I just asked my partner for his opinion on this and he said that he thinks "fewer than 1%" of Americans would realize that it means a school for teachers.
I don't know what the intuitions of native speakers outside the US might be.
Examples of US universities that used to be "Normal" schools: UCLA and the entire California State University (CSU) system came out of the "California State Normal Schools," for example. The CSU system is the largest four-year university system in the country.