While trying to connect to a SiteGround server via Putty I had the same problem. Their instructions are pretty thorough, and must work for some people, but didn't work for me.
They recommend running pageant.exe, which runs in the background. You register your key(s) with Pageant, and it's supposed to let Putty know about the keys when it tries to connect.
In a couple of places I found suggestions to specify the key directly in the Putty session definition: Putty Configuration > Connection > SSH > Auth > "Private key file for authentication", then browse to your key file in .ppk format.
Doing this without running Pageant resolved the problem for me.