Neither am I, but since PHP5 the plain old mysql functions had to be specifically downloaded and installed (with quite a bit of pain as I recall - on win32 anyhow), and mysqli and PDO (and drivers) were the bundled mechanisms - is my recollection.
To make sure, try running:
<?php
print_r (get_loaded_extensions());
on a blank page, and see if it is there.
Find a new host - they’re incompetent. They should have your account and the other sites hosted on the servers in their own VM’s, leaving you free to pick your own OS, let alone what version of PHP you want to run. They have no excuse - doing things the old way exposes your data to exploits and malware that hits any of the sites in the shared hosting solution.
A 2 year old thread, resuscitated by a (now deleted) self promotion post.
Thread closed.