Original question: yes, it is normal to be concerned. There is lots of information on sites such as AgeUK and CAB. Money Saving Expert is useful too.
It is OK to gift money to family members whilst you are hale and hearty. There are no strict guidelines around 'deprivation of assets' for care cost purposes (see above).
You can consider trusts. There are, for example, trusts you can set up and pay into for the purposes of children's education, and you may want to google those.
I understand that you can put a house in trust. I became aware of this when a distant family member, who needed significant care, had to sell the house that he (and other family members) regarded as a family home rather than 'his'. However, that was not the legal situation; and a cousin told me that it should have been put into a trust. However, that has its own implications, around who can authorise its sale, in what circumstances etc. and you need very specialist advice.
Although it can be headline grabbing, actually,most people do not end up spending enormous sums on care. I think the average stay is less than 2 years, and regular pension income funds a good bit of that. So some people are spending a lot on elaborate schemes that may backfire, in order to avoid spending a relatively small sum.