The issue is that the default graphics driver is not working correctly with your video card. The safe graphics mode disables the incompatible part of the video driver, but leaves you with no video acceleration.
As a temporary fix, you can enable safe graphics mode for the installed ubuntu by selecting an entry in the grub menu and editing it (press e) to add nomodeset
to the linux
line. See What is safe graphics mode?
Note that this only changes the in memory version of the grub menu, this is temporary only. If this doesn't work, it is possible to completely disable graphics and force text mode via a similar edit.
Once you have booted into a usable video mode (graphics or text), you then need to install a video driver appropriate for your video card. The command sudo ubuntu-drivers install
should do the hard work of selecting the correct driver and installing it. (Note that this will try to download drivers from internet, so your network must be working for this to help.)
After successfully installing the driver, you will need to reboot. If everything works, you shouldn't need to use nomodeset again.
ubuntu-drivers autoinstall