"The thing is, Putin can get away with bombing Syrian rebels because he is very effective (compared to allied sorties) also against ISIS targets."
He's barely even hit ISIS targets, he's been focussing on moderates an al Qaeda, his actions have, more than anything, helped ISIS, because they've massively damaged one of ISIS enemies - moderate rebels, and in fact, as moderate rebels have been losing ground as a result of this, ISIS have been gaining ground. Putin wants to create an Assad vs. ISIS scenario to force the West to support ISIS.
Here are some sources. This map shows the pattern of Russia's bombing campaign compared to that of the West. There are a lot of maps from a lot of sources showing the same pattern if you don't trust the BBC FWIW:
Here's a fairly detailed analysis showing the advances:
You can actually find similar evidence even on Russian news sources, although they label moderate/al Qaeda areas as ISIS, even though no one in the entire rest of the world including the people in those areas actually identify with ISIS. At best Russia's done a handful of actual strikes on ISIS to try and maintain this false credibility that it's about hitting ISIS, rather than supporting Assad and ISIS to help turn the tide in favour of Assad and force everyone to rally with him against ISIS, because the fact is, if it comes to that kind of two way fight as Putin wants, the West will have no choice but to back Assad against ISIS.
"The thing is that UK and EU public opinion is very much in favor of Russian airstrikes in Syria."
I don't think this is even remotely true. I don't know anyone in the UK or any of my friends in Europe who even remotely support Russia. Support for Russia largely exists on the fringes in groups that feel under-represented in Europe, such as far right groups like the BNP and their supporters.
"I have seen BBC News stories where the comment section was flooded with messages in support of the Russian attacks (it was a story about Turkey warning Russia not to interfere in its air space). And practically all the highest-voted comments were pro-Russian strikes."
Rather than take that as a reasonable vox pop of public opinion, you need to understand that the BBC comments section is fundamentally broken and does not even remotely represent popular opinion. It is a favour melting pot of government propagandists, and zealots. The same comments section on political stories also suggests 80% support for xenophobic and racist parties like UKIP, but actual general election day polls gave them a meagre 13%. You can't discern anything of merit from the BBC comments section, it's a cesspit of nonsense and even the moderators are part of the problem. There was a UKIPer calling Polish people a bunch of lazy spongers, and I responded saying I've actually found the Polish to be incredibly hard working, often more so than many British folk. Guess which one of us got censored and which did not? When the leanings of the moderators there (I believe the BBC contract that work out to a 3rd party company so it's not their direct staff at fault but the company they subcontract to) supports that world view, is it surprising that that view is more prominent there?
I do agree that Russia is more effective though, you're right about that, but it's largely because whilst the West at least tries (even though it regularly fails) to avoid civilian casualties, Russia just doesn't even care about them. If there's a rebel fighter in a hospital of 100 innocents, Putin is happy to see that hospital bombed to get that one fighter - he's a leader who after all openly broke the Geneva Convention and admitted to committing a war crime by removing identifying marks from his soldiers and claiming them as civilians in Crimea. Russians are effective because they have no care for the concept of collateral damage, though fail to grasp that that attitude is precisely why people might want to blow up their airliners (the US, UK et. al. have made the same mistake enough even when they unintentionally fucked up).