Here is a perfect example from a past iOS version that explains exactly how you could disable Search/Spotlight web images and then also block it with screen time. Unfortunately, it doesn't work like this anymore because Apple changed the settings and options within Screen Time for iOS 18
The issue now is Search is still visible in the root settings page. Inside of search there is an option called "Show Related Content" which is what allows the explicit web content to come up in Search/Spotlight. You can toggle this setting off, but you cannot hide that setting with Screen Time, so it does no good to just toggle it off as anyone can go back in and turn it back on.
I'm really disappointed that Apple hasn't done anything about this at this point.
For the previous iOS versions:
"To block or restrict Spotlight Search on your iPhone SE, follow these steps:
1. Disable Siri & Search Suggestions for Specific Apps:
1. Open the Settings app.
2. Scroll down and select Siri & Search.
3. You will see a list of all your apps. Tap on each app you want to restrict.
4. Toggle off the options:
- Learn from this App
- Show in Search
- Show Siri Suggestions
- Show in Look Up
- Show When Sharing
2. Disable Siri Suggestions Completely:
1. Open the Settings app.
2. Tap on Siri & Search.
3. Toggle off the following options under "Suggestions From Apple":
- Suggestions in Search
- Suggestions in Look Up
- Suggestions on Lock Screen
3. Restrict Content in Spotlight Search:
1. Open the Settings app.
2. Go to Screen Time.
3. If Screen Time is not enabled, turn it on and set a passcode.
4. Tap on Content & Privacy Restrictions.
5. Toggle on Content & Privacy Restrictions.
6. Tap on Allowed Apps.
7. Disable Siri & Dictation.
These steps will limit the content that appears in Spotlight Search and disable Siri suggestions. However, completely blocking Spotlight Search might not be possible, as it is an integral part of the iOS system."