I have an Excel spreadsheet with every major league linescore from 1901 to 2021 (imported from Retrosheet's text file logs). This is exciting news for me. (It may be somewhat less exciting for you.)
The first thing I can tell you is that in those 121 years of major league baseball (a total of 203,095 games), there have been 16 instances of a team scoring, over four consecutive innings, 1-2-3-4 runs. It has happened five times in the first four innings (what I call a Ramones linescore*), by only two franchises:
June 4, 1925 - Pirates 16 vs Phillies 3
May 29, 1943 - Pirates 12 vs Phillies 4
September 7, 1977 - Tigers 12 vs Orioles 5
September 23, 2001 - Tigers 12 at Red Sox 6
June 14, 2016 - Tigers 11 at White Sox 8No team has scored 1-2-3-4-5 runs in consecutive innings (since at least 1901) at any point in a game.