I clicked a link on twitter and it had nothing on the website and then some text appeared on it that said "your IP address has been logged to the authorities for having CSAM".
Can websites have your IP address and can websites report you to the police?
By the way, I don't, I'm just a 14-year-old on twitter.
... can websites just have your ip address
When you communicate with a website you a) need to have their IP address in order to send them your request b) they need to have your IP address in order to send you the answer back.
Directly exchanging the IP address with the website can be avoided if you have some third party in between which hides your IP address - this can be a proxy, VPN or similar technology. But then again - you need to have the IP address from this third party and they need your IP address in order to exchange data. This can be done in multiple steps (i.e. more third parties involved) for even better hiding of the IP address.
... can websites actually log you to the police?
They can but this does neither mean that they actually do nor that it has any result. Details depend on what you've done and what laws apply to you and the site. This cannot be answered here, try Law Stack Exchange instead.
If it is like you've described it probably is just an empty claim to scare you. There are known scams which then try to make you send money, call some number or similar to avoid consequences - don't fall for it.
Every website in the world will know your IP address, because it needs it to send a reply to you. The exception is when you use a VPN server, where the VPN server knows your IP address, and the website knows the VPN servers IP address. So any website can easily show a page with your IP address and some nasty message on your screen. That is absolutely normal.
Now with their sick claim of you being caught watching child porn: If you never did this, then they are obviously lying. If you did it, then this website has no chance of finding this merely by you clicking some link, so they are lying as well.
And no, they are not reporting you anywhere, because they don’t actually have the slightest clue what you are doing on your computer. If they did know, that would be proof that have broken a few laws.
PS. If you get a call from Microsoft, how do you know if it’s Microsoft or a scammer? Easy. It’s a scammer. Microsoft doesn’t call you. What if your bank’s fraud department calls you? It’s a scam. Your bank’s fraud department doesn’t call you. But what if someone is emptying your bank account and your bank tells you to open a new account and move all your money into it to keep it safe? It’s a scam. Your bank can lock your bank account down at any time without your help. And they wouldn’t ask you to move your money. If they thought that’s needed for your protection then they can move your money into a new account without your help.
