Sağlamak means to provide in modern Turkish. Sağlam means fortified in modern Turkish. However it considered to be modern evolution over the word for providing "sağla-" [*].
The word “Selam” actually originates from the word “Sağla+m” which originally meant “my support” in Turkish. (It may be Sağlağım. Not possible to assess exact pronounciation.)
It simply stand for “Do you need my help?”. Just like holding a food in your hand asking to a person. Chocolate?
Also there is a very common Turkish phase “Kolay gelsin”. Which means something like “Good luck with your work.” (Actually it emphasize ease instead of luck.) Just like saying, I can't help you but I wish you luck.
The word Islam also considered to be derived from same root as salam. Considered to be referring to peace. Because why not.