[delidumrul]

Actually, this is not an exception in Turkish. It is because of that "meyhane" is a foreign word (adapted from Persian). In any case, no one will expect smo to pronounce these 'e's distinguishably.

[patates]

First of all, nobody will understand if you pronounce it "meːɐ̯t" (as in "mehr" in German + "t").

Mert is actually also a foreign word. Well, most (I'd guess 90%) of the words starting with "me" are of Arabic or Persian origin.

But the difference is there in Turkish words as well. Geri, gerzek (which is interestingly a shortcut for "geri zeka") and gevrek.

There are many "e"s in Turkish, difference among which is very sparsely documented.

Don't even get me started on the "ğ" (which is normally supposed to just make the previous vocal letter longer but took on many other roles).

[tehlike]

We should probably say "okundugu gibi yazilan" instead of "yazildigi gibi okunan" :) That'll do.