This is complicated, to say the least. TMDb aim to follow the original seasons, and History is known to frequently re-arrange their seasons. I'll look into this more later, but the original listing might have only five seasons:

1. History first announced an eight-episode run spin-off beginning October 2010. I don't know what happened, but they only aired four episodes. No website archives. They then announced a series premiere on April 15, 2011. The first website archive was at the end of July 2011, and not until November for the episode guide. It doesn't tell us much, except that they seem to put all the episodes in the same season. In fact, all episodes from October 2010 to February 2012 seems to be part of the first season.
2. A new season began on April 11, 2012 with Blast Off!. It was clearly announced on the website, and there's a break between current and former season(s) in the episode guide. Based on the same episode guide, all episodes until January 2013 seems to be part of this season.
3. The History website got a makeover around April 2013. The new episode guide has seasons that don't seem to match previous data. It marks the beginning of a new season (Season 4 in the 2013 listing) in April 2013 .
4. According to the new episode guide, the next season (Season 5 in the 2013 listing) began in January 2014.
5. The History website got another makeover with new seasons. The next season (Season 7 in the 2016 listing) began in January 2016. I can't see the earlier seasons, but this new order might match the seasons in the History Vault https://watch.historyvault.com/shows/american-restoration