Does anyone have any sources of major news outlets reporting that the Texas gunman was Muslim? The problem is treating tweets as "news" and this is a problem by media outlets left and right. There are all kinds of tweets that are false and misleading that trend in both right and left circles why do we only complain about one side.
Major news outlets, no. Not so major outlets, sure.
All the Googling I've seen seems to point to a "Santa Monica Observer" as the primary source of this claim. (See the Snopes article: https://www.snopes.com/texas-samir-al-hajeed-sam-hyde/ ), Snopes also links to a "Freedom Daily" and there are a couple of other sketchy news websites with this claim.
The Snopes article also mentions that a comedian known for trolling-style pranks (Sam Hyde -- https://en.wikipedia.org/wiki/Sam_Hyde#Hoaxes_and_pranks ) is tangent to these claims. So my best guess is that you are correct, someone treated a trollish tweet or post as actual news.
I'm not sure why bias is brought up all the time in "fake news", because bias is not the same as fake. Nobody should have issues with conservative news that is known for better vetting, even if their bias is strong (your Wall Street Journals, National Review, etc.). Fake is fake whether it comes from the right or the left. It is true that conservative fake news was much more prevalent in the 2016 election (https://web.stanford.edu/~gentzkow/research/fakenews.pdf), but "progressive" fake news definitely exists.
The tweets exist on "both sides." Sure. But how effective and endemic are these misleading tweets in each sphere?
I'm having a real hard time thinking of analogous lies on the left that have readily observable support or momentum in comparison to misinformation like the origin, religion, or race of a mass murder and/or claiming that an elected president wasn't actually born in the U.S.
Hard to dismiss such misinformation that's heartily consumed and propagated in whatever sphere as just being "tweets" or saying "it exists over there too!" when the effects of said misinformation in each sphere are significantly different.