People make a lot of assumptions about what it means to “major/minor” in music, so I think it might help to clarify.
It’s mostly about what degree you decide to pursue: a Bachelor of Music (BM), Bachelor of Arts (BA), or Bachelor of Fine Arts (BFA) are generally your choices. These degrees mostly differ in their course requirements, e.g. percentage of music courses, core academics, electives, etc. Some music schools also offer a Bachelor of Science (BS) for music technology or business fields.
There is no legal/credential difference between a “conservatory” and a “school of music”. Oberlin calls their school of music “Oberlin Conservatory of Music” while at University of Rochester, it’s “Eastman School of Music”, and Berklee, which is often cited as an example of a conservatory is actually named “Berklee College of Music.”
There is also no rule about which of the BM/BA/BFA degrees might be offered by conservatories or stand-alone music schools in a University. Many music schools offer more than one. Many music schools define “majors” as specific focus within music, such as “Jazz Studies”. There maybe some surprises – Berklee doesn’t have a “Jazz Studies” major, for instance, and The New School’s Jazz offers a BFA but not a BM program.
I’m assuming you’re asking about schools that have music departments but not stand-alone music schools. Columbia, Harvard, Yale all have music departments with excellent jazz faculty. University of Pittsburgh has a good jazz department; they unfortunately lost Geri Allen recently. City College in NYC mentioned above has an excellent jazz program, but this is a BFA program, not BA.
In or near NYC, the schools most often mentioned for jazz are: Manhattan School of Music (MSM), Steinhardt (NYU), City College, The New School, Juilliard, Columbia, SUNY Purchase, William Paterson (NJ).