A small number of winds is good, because there's less string to stretch, but there have to be enough winds to hold the strings securely to the posts.
From your pic (nice and clear!) it does look like there aren't enough winds. The usual method with those vintage-style tuners is to pull the string tight, cut it about 2 inches past the tuner post then push the end into the centre hole of the post and start winding. You should have about 3 full windings round the post for tuning not to slip.