Here, we not only have to have similar amounts of units to counter single-unit spam, but nobody can agree on WHAT UNIT TO USE. Which makes me, as I stated before, very hesitant about thinking that anyone has any idea what they're talking about.
It sounds like you aren't even trying what people are suggesting and instead just pouting about your loss. It was mentioned several times and if you seriously tried it, you'd see that it works. Flak ships and/or fighers - not bombers.
There's no disagreement on what to use. Some are saying fighters, others are saying flak. They're both right. I use them all the time against TEC LRM spammers and Advent Illum spammers, and honestly, I have yet to lose unless I'm messing around testing something. Most people, only knowing the lrm spam tactic, quit when they find their assault's not nearly as potent as they think it is.
It's true that against mass illums, it's a bit tricky because illums do 100% dmg to flak just as flak deals 100% to illums. However, pound for pound, without external help, flaks will still win. If they have support, you may very well lose. Thus, you need to counter their support with yours or simply take out their support ships as quickly as you can.
Needless to say, rts games are all about micro. Move your wounded ships "through" them to force them to turn around to shoot the focus fired ships. Of course, they'll still hit you with the other beams, but they'll only be shooting 2/3 beams once their target is destroyed. This trick is even easier against TEC and Vasari LRMs since they completely stop shooting and need to turn around to face the rest of your fleet.
If this is too hard for you, you can simply take your entire flak fleet and drive it "through" the enemy lines back and fourth. this will force the enemy fleet to keep turning around to shoot meaning you'll only really be getting hit by their secondary beams, whereas the flak ships' offensive ability won't be that hampered.