Glad to say Supergirl fans are generally not guilty of this. To date I have almost not seen a Supergirl fan hating on the other shows.
But, if you scan social media, for example on Facebook, Twitter or Youtube (Youtube being actually a lot more harmonious lately), you can see the violent backlash against the increasing integration of the CW superhero shows.
At the frontline are the people that watch Arrow only. I get it, it was the first DC show on the CW, and it's darker than the rest. That somehow makes certain Arrow fans feel a special entitlement. They are the most vocal in beating down the other shows, the magnitude increasing in order of LoT, The Flash, then Supergirl. The recent Musical is a particularly favourite target of theirs, the epitome of the perceived negative qualities in their eyes, which I can only guess to be light-hearted environment in a show. And the fact that Arrow is increasingly crossing over further ticks them off.
A particular photo of Melbs and Grant having a laugh at some Arrow related joke at Paleyfest incensed them. A storm of vile toxic insults against the Supergirl and Flash leads ensued on Facebook.
The integration of the multiverse is a major cause. I for one am loving the hell out of every sign of cooperation between the shows, but to some fans that watch one show only, they get really salty. A large amount of complaints are about having to watch other shows, and CW's increased cross-promotion means on every ad they post there's bound to be such hatred, normally as the top voted comments.
Less so than Arrow, but there are those who only watch The Flash and they also hate on everything else. They are a bit more justified though, because The Flash is a damn good show, and although I personally like Supergirl probably slightly more, rationally speaking it probably is the most well made in the Arrowverse. Still doesn't give them the right to shoot everything else down though.
Lastly there's those who don't quite see Supergirl as a worthy addition into the Arrowverse, and hate on this show only. They too splinter the unity of the DCTV fandom. Some are fueled by a rejection of a progressive feminist show?(My guess, take with a grain of salt), others possibly more justified because of the various plot faults evident currently. About the second accusation there's not much I can say in defense, I love this show but Season 3 does need to fix a good amount of things. Despite that, per each episode, I think SG is the most interesting at the moment, on par with LoT.
The impact of all this? Every time I want to look at the comments I end up emotionally slapping myself. It does not feel good to see the things you like attacked over and over. Other fans who like me, thoroughly enjoy the entirety of Arrowverse, may feel the same way.