There is more land in the northern hemisphere, more ocean in the Southern Hemisphere. Summers get hotter in the north, winters get colder. This is because land does not store/release nearly as much heat as ocean.
That's not just you. Every traditional calendar says the same thing.
As far as I can tell, the idea that winter starts on the solstice, when it's been winter for more than a month (unless you're in the tropics), is an invention of Hallmark or some other calendar manufacturer. It is based on nothing.
I tried reading Poor Richard's Almanac to see if it indicated the "beginning of [any season]", but if it does I didn't understand. (There is a column of "remarkable days", but it is often difficult to understand the entries in that column.)
Careful track is kept of the exact time of sunrise and sunset, and of what kind of weather to expect ("rain"; "snow"), but the seasons don't seem to be labeled explicitly.
https://en.wikipedia.org/wiki/Adoption_of_the_Gregorian_cale...
"Cultural influences such as Christmas creep may have led to the winter season being perceived as beginning earlier in recent years"
I think I've always considered November to be winter, and at some point I picked up that Christmas is in the middle/even towards the end to break it up and give everyone something to look forward to, so I'm always a bit surprised seeing talk of winter in the future tense once I've put on the winter duvet/started wearing a coat all the time/turned on the central heating (not this year).
Equally anything in Feb seems firmly Spring.
But I'm not trying to force this on anyone!