Unfortunately, the MinGW-w64 installer you used sometimes has this issue. I myself am not sure about why this happens (I think it has something to do with Sourceforge URL redirection or whatever that the installer currently can't handle properly enough).

Anyways, if you're already planning on using MSYS2, there's no need for that installer.

1. Download MSYS2 from this page (choose 32 or 64-bit according to what version of Windows you are going to use it on, not what kind of executables you want to build, both versions can build both 32 and 64-bit binaries).

2. After the install completes, click on the newly created "MSYS2 Shell" option under either MSYS2 64-bit or MSYS2 32-bit in the Start menu. Update MSYS2 according to the wiki (although I just do a pacman -Syu, ignore all errors and close the window and open a new one, this is not recommended and you should do what the wiki page says).

3. Install a toolchain

   a) for 32-bit:

   pacman -S mingw-w64-i686-gcc
   

   b) for 64-bit:

   pacman -S mingw-w64-x86_64-gcc
   

4. install any libraries/tools you may need. You can search the repositories by doing

   pacman -Ss name_of_something_i_want_to_install
   

   e.g.

   pacman -Ss gsl
   

   and install using

   pacman -S package_name_of_something_i_want_to_install
   

   e.g.

   pacman -S minw-w64-x86_64-gsl
   

   and from then on the GSL library is automatically found by your MinGW-w64 64-bit compiler!

5. Open a MinGW-w64 shell:

   a) To build 32-bit things, open the "MinGW-w64 32-bit Shell"

   b) To build 64-bit things, open the "MinGW-w64 64-bit Shell"

6. Verify that the compiler is working by doing

   gcc -v
   

If you want to use the toolchains (with installed libraries) outside of the MSYS2 environment, all you need to do is add <MSYS2 root>/mingw32/bin or <MSYS2 root>/mingw64/bin to your PATH.