A simple / basic answer, to provide an Upper Bound, is to use the scales
4 times
First, label our unknown balls "A", "B", "C", and "D". Next
Weigh them in pairs, and write down the winning pairs:
A&B v C&D
A&C v B&D
A&D v B&C
If, for demonstrative purposes, we assume that A=1, B=2, C=3 and D=4, then you would get this table:
1&2 = 3 v 3&4 = 7
1&3 = 4 v 2&4 = 6
1&4 = 5 v 2&3 = 5 : If we get the other 2 first, then we can skip this one!
We can then tally up each ball by "Heavy" versus "Light"
1: Light / Light / Draw
2: Heavy / Light / Draw
3: Heavy / Light / Draw
4: Heavy / Heavy / Draw
This means that our final step is
Compare the 2 balls with 1 "Heavy" and 1 "Light", to work out which is 2 and which is 3