I am looking for a word or expression to mean running out of somewhere very fast because of being terrified of something, or to avoid meeting someone.
Here's a handy single word: flee. You may flee a place or a person or a threat.
Since you are looking also for an 'expression', I would suggest:
like a bat out of hell
if you go somewhere like a bat out of hell, you go very fast He ran out of the building like a bat out of hell.
(Ref. Cambridge Idioms Dictionary, 2nd ed. Copyright © Cambridge University Press 2006.)
In Northern English (e.g. Manchester, Liverpool), this phrase is quite common
When those guys showed up, we legged it.
In US informal English there's also "skedaddle":
When those guys showed up, we skedaddled outta there!
Perhaps more commonly you can always use an intensifier to impress upon the listener that you got out of there quickly:
When those guys showed up, we got the hell outta there!
It always sounds a bit "slangy" to me, but a nice word for OP's context is...
skedaddle - to depart quickly or hurriedly; run away.
...another one I like (esp. US usage, also somewhat informal, as are most apart from flee) is...
Similar terms include scarper, leg it, beat it, clear off.
You could use the verb bolt. NOAD defines it like this:
As a side note, in recent years, this option has been a very handy word for sports headlines.
"Skip out" can be used in the said contexts as in
John was arrested July 22 after he skipped out on a $300 bill earlier at a local bar and ran away.
Macmillan dictionary defines it as a phrasal verb which means
to leave a place quickly, usually because you do not want to be noticed or want to avoid paying money
There's the expression run off, as in:
"Run off to somewhere else!"