I see... Ubuntu and its derivatives like Mint uses a different installer.
Before to go to grub you see some info, that's fine this is your kernel that is said something, than you see grub, but after you don't see anything? Is that right?
Which is the mobile GPU of this Laptop? Which is the model of your laptop?
While the general assumption that old hardware is well supported by Linux is true it doesn't apply at all to the Laptops. A lot of laptops are assembled with exotic and specific hardware that never see a floss drivers and sometimes can be very hard make these chips working properly.
Without any information your issue might be a "modeset" mis-configuration, however the parameters apply based on you hardware and without knowing it is hard to lead you somewhere...
By the way you can take a look at these page regarding your hardware compatibility with the Linux Kernel:
https://www.linux-on-laptops.com/https://www.linux.com/news/my-hardware- ... -out-here/