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I installed Xubuntu 18.04 on my 32 GB USB stick by choosing USB as a partition instead of the drives on my HDD. It took a long time to install (around 2 hours), but at the end it said that the installation was successful and to restart my PC.

Now after I restarted my PC, it wouldn't boot to my USB with Xubuntu on it.

When I view them from Windows, my USB drive appears as these two icons:

https://prnt.sc/mq1u1m

There are two of them because I used some space for swap area.

Is there anything i can do to make this bootable or should I reinstall? If I need to reinstall, is there anything I should do differently?

First, does system boot from ANY USB stick? Did you install the system from another stick or from a real DVD? If it was a stick, does it still work? Maybe booting from an USB drive has been switched off somehow.

Second, did you select your USB drive for installing the bootloader, or for those partitions only?

You can probably boot from the installation media again, plug your Xubuntu stick and install Grub manually. The full answer is here: How can I repair grub? (How to get Ubuntu back after installing Windows?) but I prefer a short way (it works for default installation parameters). You need to know the device name of your Xubuntu USB drive (/dev/sdb, /dev/sdc etc) and it's auto-mounted path (something like /media/username/USBStickname). File manager usually auto-open the connected USB drive, showing you this path. Run sudo grub-install --root-directory=/media/what-is-the-path-to-your-USB-drive /dev/sd? (with no digits. Only sdb, sdc etc, the exact name of the USB device itself). That'll fix the missing Grub.

Also, next time care about the flash memory wear: New Lubuntu install - 2GB RAM, 32GB SSD

Welcome to askubuntu!

In general, I wouldn't advise installing a full operating system to your flash drive as if it were a hard drive. The main issue with that has to do with performance. You noticed it took forever because it's not intended to be used that way.

What I would recommend is creating a live USB (like you would use for an installation) with persistence. This allows you to use a live system optimized for use on a USB while creating space on the drive to keep your files without wiping the drive every time you shut down. Since you're trying to use a 32 GB USB drive, I'm guessing you'd like to have a good amount of space for persistent files. There is an excellent guide on how to do this provided by users on this site!