Birthday can be a really hard word  for my students, let's break it down.  

B consonant, that's pretty easy but the  /ɜɹ/ vowel, that can be tricky. Bir,  

just make the r sound. Don't  try to make a vowel. B-ir, Bir, 

Brrrr! Bir,  

birth. And we have the unvoiced th,  

your tongue tip has to peek just through but don't  hold it, don't clench. Let the air flow freely. 

th-, birth, birth. Birthday. 

And day is not the problem  part of that word. Birth-day. 

Birthday.

Happy Birthday.