Police can see Google searches if the police have a warrant or subpoena and submit that warrant or subpoena to Google to access your searches.
Or if Google detects illegal images on your Gmail account or in your Google drive or Google account through automatic systems, then Google can and often will report your account to law enforcement and they will then look into your searches and data and if they find something illegal they will show up at your door eventually with a warrant to seize your devices, search your home and possibly arrest you.
You can get into trouble for googling illegal things, although it will depend on what you're actually searching that is illegal.
Simply googling some illegal things won't always get you into trouble.
Although if you google something like Child Porn and your intent is to find and look at and save that child porn then you can get into trouble.
Google also has systems in place that can automatically detect potential child porn content even when it's sent through their Gmail service.
If google detects child porn or possible child porn being sent through your Gmail account or you save child porn or trade child porn or other illegal things on your google account including google drive account they will contact authorities.
Although simply Googling the term child pornography is not illegal in itself.
As someone searching the term child pornography may be simply trying to find out about how to report the child porn or looking on the law about it.
Also while viewing and or possessing child porn is also illegal, you won't always get into trouble if you happen to stumble upon child porn.
There has to be intent.
Also searching for tools or even guides for engaging in cybercrimes such as hacking is also illegal.
And even if your search intent is not malicious, searching for topics like how to make a homemade bomb, etc can also trigger an investigation and can place you on a watch list.
Police can also obtain a warrant to access your search history from Google or your ISP and they can provide that even if you delete your search history or destroy your device or computer.
Some keywords and phrases can also sometimes trigger automatic flags from law enforcement in some cases.