I do accept that they are accusative. That is why I call them exceptions

"istisnalar kaideyi bozmaz."

Syntactically, there must be two bir şey´s.

1. ´a thing´  -  a usual noun. You can see it take the accusative ending but in that case bir should probably be interpreted as a numeral:

                     Bir şeyi de bil.

                     Bir arkadaşı bekliyor.

2. ´something´  -  an indefinite pronoun. Cannot be definite (and take accusative ending) of course because of its meaning. bir is no longer understood as an article but a part of that word. That is why in this meaning you very often find it written as one word. And there can be an indefinite article and plural ending in the same phrase like

                     Bir şeyler yazmalıyım.

I am just speculating. Correct me.

yes, I think it is something like that.

...and what makes it complicated is that both bir and şey are not stable but on their way somewhere, into more abstract and grammatical meanings, bir from a numeral to an indefinite article and şey from a noun into a pronoun. Together they make interesting combinations.

Exception of what?

We know what nominative case means and we know what accusative case means.

Bir şey anlamadım = I didn´t understand anything

Bir şeyi anlamadım = I didn´t understand one thing (among some others) = I understood everything except for this thing.

Here accusative points out to something specific.

Bir şey anladım = I have understood something = I have come come to a conclusion about something

Bir şeyi anladım = I have undertood one thing (among some others) = I haven´t understood anything except for this thing.

Here accusative points out to something specific.

Bir şey duymadım = I didn´t hear anything

Bir şeyi duymadım = I didn´t hear one thing (among some others) = I heard everything except for this thing.

Here accusative points out to something specific.

Bir şey söylemedim = I didn´t say anything

Bir şeyi söylemedim = I didn´t say one thing (among some others) = I said everything except for this thing.

Here accusative points out to something specific.

etc.