Learn Hardcore Turkish: Telefonumda bildirim var. - There is a notification on my phone.

Var: neutral, everyday “there is/are.”
Mevcut: formal “available/present.” E.g., Bildirim mevcut (formal).
Bulunuyor: formal/“is found/located,” sometimes suggests ongoing state. For phones, bildirim var is the most natural.

a/e choice: depends on the last vowel of the word (back vowel → -da, front vowel → -de). In telefonum, the last vowel is u (back), so choose -da.
d/t choice: depends on the last consonant’s voicing (voiced → d, voiceless → t). Telefonum ends with m (voiced), so use -da. So: telefon + um (my) + da (in/on) → telefonumda = “on/in my phone.”

Why is there no article like “a” before bildirim?

Turkish has no articles. A bare singular noun in an existential sentence with var is understood as indefinite (“a notification”). If you want to explicitly mark “a/one,” you can add bir: Telefonumda bir bildirim var.

When should I include bir?

To emphasize “one” or to make the indefiniteness clearer: Telefonumda bir bildirim var.
Without bir, it’s still indefinite and very natural: Telefonumda bildirim var.
If you mean the number 1 (as opposed to 2+), bir is recommended.

Should I pluralize bildirim if there are many?