Either your email server or your email app has a limit of displaying 5,000 emails. Once you figure out which one it is (try reading the same account in a Windows email program like Thunderbird), you'll know what needs changing. If it's the server, and it's not your company's server, there's nothing you can do about it except archive the last 4,900 emails and keep archiving every time you get past about 4,000. If the server owned by someone else limits you to 5,000 non-archived emails, that's what you're limited to. (If you need to be able to read any of more than 5,000 emails at once, you're the fastest reader in the world, and your business model is completely wrong. If not, archive at the end of every month, and read whichever archive you need at the moment. The company may have to set up a server just for archiving your emails, if you get so many of them. There are 10TB drives available. If you need to archive more emails than that, why are you here? General Motors has its own tech department. [You'd have to be running a company at least that size to need to archive more than 10TB of email, or you'd have to be running a cloud server farm.])
Oh, and ask whoever does the hiring to hire techs for your tech department - promoting mailroom clerks to "technician" doesn't do the company any good. (This problem is IT kindergarten, it's not even IT 101.) If you have an IT department - a real one, not just clerks working in a room with "IT" on the door - you should almost never have to seek outside help, and when you do it should be something almost no one here could help you with. Things that have to be analyzed on-site, for instance. Or dual-homing your web server and phone lines over split-direction fiber, so that when the idiots redoing the building next door Ditch Witch through your fiber, you don't know it until someone sees the orange plastic thing sticking up out of the ground.
Didn't the IT person who administers your email server even have a clue about any of this?