Say what you will, but using "google" is just the fastest way to say "search the web." Unless a better, equally convenient phrase comes along, it’s not going anywhere anytime soon.
I can't think how anyone is going to be confused and think I want them to check the bins or something, but maybe we search for different things on the web.
"Hmm. Not sure. Maybe google it?" is pretty innocuous no matter the context.
"Hmm. Not sure. Maybe look it up?" can sound dismissive or condescending in some cases to some people's ear.
It's a weird quirk of English where being non-specific is like saying "I don't know, you figure it out", where being very specific is like saying "Here's the solution (that I would use)".
I'm still not convinced this is quite as ergonomic as it could be though because of a quirk of phrasing; instead of saying "I googled that", you'd need to say "I looked that up". As someone who often struggles to enunciate super well most of the time without my speech becoming slow to the point of tedious, it does still feel a bit more verbose because "googled" is such an awkward word that you can sort of pronounce it almost like 1.5 syllables by softening the second 'g' to produce something like "I goo-ulled that" and its still is intelligible. It still might be close enough that it can catch on, and I'm not at all opposed to it because of this minor inconvenience, but I do think that there are factors beyond a pure syllable count come into play even just in terms of linguistics rather than social convention.
To google (two syllables, only slightly glibber, due to no "b-s" plosive-fricative combo).
Maybe "web search" can be shortened to "bsearch" or "besearch", like "web log" got shortened to "blog".
"besearch" has grammatical precedence: bestow, bequeath, behead, bewitch, belabor, befriend, ...
So Google is polluting what it means anyway.