The reality is far worse than this. You need to go to the Original Comiskey Park, a pitchers park, and a season before most of us (I was conceived during this season) were even born, to reach this level of run scoring futility.
1968 was the last season the White Sox did not score 5 or more runs in 10 or more games after 37 games.
There was no designated hitter at the time, and this was the final season before MLB lowered the pitchers mound and shrunk the strike zone to the current dimensions.
https://www.baseball-reference.com/teams/CHW/1968-schedule-scores.shtml
The White Sox were so bad this season,
A. The White Sox split home games between Comiskey Park and Milwaukee County Stadium, in an attempt to draw more fans. They drew 803,775 (9th out of 10 AL teams) over 81 home games, or less than 10,000 per game.
B. GM Edwin Short went through three different managers this season, if only we could be so lucky.
Eddie Stanky (34-45)
Les Moss (12-24)
Al Lopez (21-26)
C. The team finished with a 67-95 record.
D. It took until the 53rd game of the season for the Sox to score 5 + runs for the eighth time, a 5-2 win over the Yankees at OG Yankees Stadium, with Wilbur Wood getting his 4th win of the season in front of 5,340 fans (yikes).
The 2022 White Sox will need to go another 16 games to tie, or 17 games with less than 5 runs each game to pass the offensive futility of the 1968 Chicago White Sox, a team handicapped by both the pitcher hitting and a higher pitchers mound.