I agree - there are problems from all extremists in this issue. I despise political extremism in all its ghastly manifestations.
As you rightly point out, the word "liberal" has been well and truly highjacked. I consider myself "liberal" in the traditional sense of the word, but absolutely not in the professionally offended authoritarian activist sense of the word spouted by the right. And speaking of the right, history teaches fascists gain power when they are able to scapegoat the "left" (whatever the hell that means these days) in such respects, and dumb down discussion to stupid for-us-or-against-us polarisation. The problem is the "left" (for want of a better term, as again, what I mean by "left" has been rather lost in the mix) respond with their own polarised idiocy, leaving centrists like me getting it in the neck from both sides. I won't lie, it really pisses me off that being a sane, rational, humane indidual who wants to evaluate the points from both sides is seen as somehow a vice these days.
For example, I am not anti-capitalist, but you'd have to be a moral imbecile to not see how the poverty gap urgently needs addressing. So yes, I'm all in favour of properly funded state healthcare, education, social services, pensions, nationalised essential utilities (water, electricity, gas, railways, etc) and that those most able to pay ought to pay more in tax. On the other hand, I despise the short-sighted inability of certain lobby groups on the so-called left, who determinedly refuse to understand the difference between depicting something and endorsing something (idiotic attempts at banning To Kill a Mockingbird in US schools for example, though the right are just as ridiculous with book bans of their own) who wish to act as thought police with their ever evolving definition of what exactly constitutes "hate speech". But there is a clear, demonstrable difference between expressing an opinion (albeit one you might hate) and inciting violence against a minority group. Example: Criticising the religion of Islam is not the same as urging racist attacks on Muslims, mosques, etc. No one has the right not to be offended.
Rant over. (I'm from the UK by the way, but my comments apply here as well.)