Apologies I wasn't clear enough…
Drag a link from Safari to the Desktop (forget about those springs in the Dock - they are useless for editing they look 'real' but that is a lie). You will have to re-create them all I'm afraid.
Get info on the new weblink, click the icon, so it highlights ready to drop an image onto it.

Find the image you like & drop it onto the icon…

Now add that to the Dock & start cursing about how Macs are simple - why it has to be this way I do not know 🙂
The next piece of bad news is that if you delete those new weblinks, they will stop opening from the Dock, move them to a folder for safekeeping.
All of this makes me fee that you are doing something wrong - generally when things are this clunky Apple didn't expect this use case & be prepared for them to break it in future, sorry.
Next step - wonder why screenshots need massive furry shadows 🙂
P.S. some images fail to look good as icons, transparency can make things fall apart or sometimes you get generic icons instead.
Good luck