I think it began as a horse race in 2005 when the companies began to improve the performance of the bland LCD displays. Sony was good, but always cost a little more and I think it hurt them in sales. LG had a very good picture initially, and Samsung was a close second, followed by Toshiba and Sharp. All sets had problems with clouding, flashlights, smearing, and, in 2007, features became available to improve motion. Samsung stumbled with automotion plus which created the tri-ball effect, in which a forward pass displayed three footballs. Toshiba had a set that was called the green monster, because the green couldn't be toned down. Sharp had a problem I can't recall, only the refrain, friends don't let friends buy Sharp.
As the years passed, I think Samsung must have had more resources to devote, because they developed sets with a gorgeous picture, flooded the market with new (sometimes buggy) features, a wide variety of models, and regularly released new models twice a year. LG played it conservatively in terms of features and models, and the other brands didn't seem to keep up. Currently, Display Search reports Samsung has 28% of the market followed by LG at 15%, and Sony at 8%, followed by all the rest, none higher than 6%.
During this time, Toshiba was also developing computers, which the others weren't into yet, and which have a very good reputation. Possibly they didn't have as many resources to devote to TVs.