"Lest" or "Or?"

Colloquially, I would always use or where I would formally use lest. For example, "go to sleep, or you'll be tired" versus "go to sleep, lest you be tired."

Has this usage of or been around for a while, or have people phrased sentences like this differently? Perhaps "If you don't go to sleep, you'll be tired."? Is it acceptable to use or this way in formal English?

'Lest' is an old word - OED citations begin around 1000AD. It is a more concise way of saying "to avoid the possibility that". As such it is still a perfectly good word introducing a clause expressive of something to be prevented or guarded against, although this fineness of expression is not commonly communicated in this way today.

Ordinary speech today often uses a less nuanced vocabulary and communicates the additional meaning using prosody.

"Go to sleep, lest you be tired" - is a suggestion that you should try to sleep because otherwise it is likely you will be tired.

"Go to sleep, or you'll be tired" - is an unambiguous assertion. Unless you go to sleep you will be tired.

And, of course, symbols and places of remembrance often explain their purpose, explicitly, by saying 'Lest we forget'.

I think of 'lest' as a form of unless. In your example, "lest"would mean "unless you want to be tired'. 'Or'in this case means the words after it describe a definite consequence of not going to sleep.