gcc -g -c -lpthread -lm test.c test1.c
gcc -g -c test3.c `pkg-config --cflags --libs gtk+-2.0`
gcc -o output -g -lpthread -lm test1.o test2.o test3.o `pkg-config --cflags --libs gtk+-2.0`

I am getting for the first step:
gcc: -lpthread: linker input file unused because linking not done
gcc: -lm: linker input file unused because linking not done

and getting:
gcc: -lgtk-x11-2.0: linker input file unused because linking not done
gcc: -lgdk-x11-2.0: linker input file unused because linking not done
gcc: -latk-1.0: linker input file unused because linking not done
gcc: -lgdk_pixbuf-2.0: linker input file unused because linking not done
gcc: -lm: linker input file unused because linking not done
gcc: -lpangoxft-1.0: linker input file unused because linking not done
gcc: -lpangox-1.0: linker input file unused because linking not done
gcc: -lpango-1.0: linker input file unused because linking not done
gcc: -lgobject-2.0: linker input file unused because linking not done
gcc: -lgmodule-2.0: linker input file unused because linking not done
gcc: -ldl: linker input file unused because linking not done
gcc: -lglib-2.0: linker input file unused because linking not done

If I remove -lpthread for example in the first step. Then it can't compile. Any way to resolve this?

Thanks

Remove the "-c"

The man page for gcc says:

If I compiled without -c, then it gives an error:

/tmp/ccsKpvMq.o(.text+0x11b7): In function `log_msg':
/home/test/project/test1.c:453: undefined reference to `message'

message function is found in test2.c with header test2.h already included in test1.c

Could you post test1.h, test2.h completely, and test1/2.c up to around the first function (that is, only the includes, declarations, etc.)?

You need to list all the .c files on the command line that contain code your program needs.

If test1.c calls a function in test2.c, then including test2.h is not enough. You need to execute this command to compile it:
#include'ing the header only informs test1.c that the function exists. The compiler doesn't automatically look at test2.c even though test1.c includes its header file.