This question points to a competency problem on your end - structural. I would review the runbook and SOPs.
I have school management system
That is custom software, I can only assume, because you do not name a product. That you got programmed. Somehow (as in: you obviously are over your head competence wise). The programming did NOT prepare for this scenario, so your programmers must fix it.
I tried to point school1.com to url(school1.mysite.com) and it worked well for
homepage, but when I navigated to another page by clicking some link I
have been noticed that url is still school1.com actually
school1.mysite.com if displayed inside iframe.
Well, given that is your software you should not try it - but test it. In a test environment.
The obvious problem here is that links between pages seem to be hardcoded in some way - relative links (href = "/blabla") will never resovle to a different host name. Non-relative links only MUST be used when changing the domain (switching schools or to an admin site), but can be avoided within a site - which is also saving some bandwidth. Somehow your programming does not use relative URL's, and does not account to the fact that a custom domain shall be used.
Classical failure.
Either from your programmers, or your support (which granted a not-implemented feature to a customer and now has sold a not supported configuration). Would not be the first time I see this ;)
Whatever you do, someone needs to start looking into the code and fix it and make a custom domain a supported scenario.
You COULD use a proxy under the new domain, including http text rewriting (changing all links in the pages), but that is like a very complex hack on top of a broken/not supported configuration.
May I ask why this question even ends up here? This is a classical scenario where I would expect the lead developer to step in. He should be very aware of compromises made and inherent limitations - especially of something as fundamental as domain name handling in the code. Whoever tried to set this up should have realized this is not a normal setup and not covered by their runbook. And as it is not covered and obviously a deviation, it should have resulted in reading documentation and then (as it is not supported there) an escalation. To said lead developer.
