Set up your account as IMAP on both, PC and Laptop. That will give you always the same view of your emails regardless of which device is used.

How do I set up the accounts as IMAP

First of all, check how it is currently set up.

Tools|Account Settings|{select account}|Server Settings

What does it say at the top, next to "Server Type"?

It says POP mail server

On the second pc, install and configure Thunderbird with ALL of those presented settings on that tab, this will pull in all the same info.

POP? That is bad news.

Left to itself, Thunderbird will choose IMAP. If it gave you a POP connection then that says that your email provider either doesn't offer IMAP, or chooses not to advertise it.

If you were offered a choice and specifically advised to select POP (do you remember doing anything like this?) then I'd have to ask what motivated that decision.