This is not consecutive matches, just the winning matches.
Guess what? The record is 6!
Glenelg in Round 20 1970, 1st Semi and Preliminary of that year, Rounds 5, 8 and 9 1971 were all bigger wins than the last. Starting with Round 19 1970 (5 points) then 9, 16, 18, 37, 46 and 52.
Sturt from Rounds 4 and 16 1992, Rounds 15, 17, 22 and 23 1993 had margins of the following: (with Round 11 1991 being a win by 2 points) 10, 11, 19, 29, 53 and 57 points.