Since replicating this problem on other systems running macOS High Sierra 10.13.4 has proven difficult, it's probably worth starting with some basics to see if we can resolve or narrow down the problem.
Your question doesn't actually state what you've already tried (if anything) to resolve this, so perhaps you could start by booting into Safe Mode to see if that makes any difference.
Boot into Safe Mode
Follow these steps to boot your Mac into Safe Mode:
- Fully shut down your Mac
- Restart your Mac
- Immediately press the Shift key and keep it down
- Let go of the Shift key when you see the login window (NOTE: If you have FileVault enabled you may need to log in twice).
- Take a note of what happens (i.e. does shift+control+command+4 work for pasting screen grabs?)
- Exit Safe Mode by restarting your Mac as normal
- Test again
Once you've booted into Safe Mode, let me know how you went and we'll go from there.
Okay, since booting into Safe Mode made no difference, we should now try resetting the NVRAM. Since you haven't specified the model of Mac you're using, refer to Apple's How to reset NVRAM or PRAM on your Mac page.
[EDIT] - Known bug in Google Chrome
Not really an answer - at least not the one you're looking for - but I decided to check https://bugs.chromium.org to see if this was a known issue. It appears a number of issues relating to copy and paste (and the clipboard generally) in Chrome for macOS have been raised by users. In some cases the issues affect only text, while in others images and text. Some of the reported issues affect Windows, Linux, etc.
Below are some examples:
It's hard to know whether any of these are related to your issue or not, especially since not everyone can reproduce it, but it could also be an intermittent bug or combination of bugs.