I started by downloading each of the dependencies as a .tar.xz file
You did not specify from where you downloaded this file, but the fact that you got a .tar.xz file implies that you did not use the package manager. For Linux, it is a good idea to use your distribution's repositories (unless you need a library version not found there). It removes much of the guesswork.
Linux Mint provides three interfaces to the package manager: Software Manager, Synaptic Package Manager, and the command line. In my experience, Synaptic Package Manager is best for finding development libraries. Search for and install the package libgtk-4-dev. (If you cannot find the package in the graphical interface, try the command line interface; in a terminal, type sudo apt install libgtk-4-dev.) This should give you all the dependencies as well.
To point your IDE to the installed packages, use pkg-config. The necessary compiler flags are produced by
`pkg-config gtk-4-1 --cflags`
and the linker flags by
`pkg-config gtk-4-1 --libs`
Be sure to type (or copy) the back-ticks (`), as those are significant.
For Code::Blocks in particular, the project build settings are under the "Project" menu, "Build options...". The compiler flags go in the "Compiler settings" tab, "Other compiler options" sub-tab. The linker flags go in the "Linker settings" tab, "Other linker options" text entry box.
Disclaimer: package names can very, but these names should be correct for Linux Mint, Debian, and Ubuntu. Other distributions might use other names. For example, the Fedora package is gtk4-devel, which becomes gtk4 when getting the flags.
If you really, really want to make your download work, we'd need to know which file you downloaded. But I think you're more interested in getting GTK working.
Since you're using GTK and C++, let me point out that gtkmm is "the official C++ interface for the popular GUI library GTK." The package name in Linux Mint 22+ is libgtkmm-4.0-dev. (It looks like the official repositories for older versions of Linux Mint only have version 3, libgtkmm-3.0-dev.) Drop the "lib" and "-dev" for the pkg-config commands, leaving gtkmm-4.0.
.libfiles are windows specific, this tutorial is pretty much useless on linux. on linux you first download the dependencies using the system's package manager (apt on mint), then just link to it using your build system, gtk uses meson so ... ditch code blocks for gtk builder, i think it also allows you to use snaps which simplify packaging. or FWIW you can go for vcpkg + cmake as alternative to apt + meson but the steps are a bit more involved.libgtkand end with-dev. Also, as IIRC Mint is Debian-based, so you could look for tutorials on how to use these libraries with Debian. And I'm quite frankly surprised that the official Gtk documentation doesn't have a few notes about that.