If you get a package through USPS and it's too big for your mailbox then what will happen to the package depends on where and how you get your mail.
If you get your mail at a post office box inside a post office and the package is too big for your mailbox then they will either put a key in your mailbox and put your package in a parcel locker.
Or they will leave a yellow slip in your mailbox to alert you to pick the package up inside the post office.
If you get your USPS mail delivered to your home at a mailbox on the street then they will either leave the package on your doorstep or they will drop a yellow slip in your mailbox telling you to come to the post office to get the package.
Most times though the package will just be left at your doorstep.