pronunciation - Pronouncing the "th" sound in American accent

First of all, I'm not a native speaker (I'm a Vietnamese) and I'm still learning English as my university major (using American accent, mostly) so I can't really say I'm as fluent as a native speaker.

However, I can say I have no problem pronouncing the voiceless "th" (/θ/) as taught by many sources including my lecturers by placing the tip of your tongue between your teeth but just blow air through your mouth without vibrating your vocal cords. My pronunciation of, say "think", always sounds like a little bit of air at the beginning (when I place my finger in front of my mouth, I can feel the air blown out), followed by the sound of "ink".

So here's the thing I find irritating: when watching American TV shows, movies and especially listening to pop music, I usually (if not always) hear they pronounce/enunciate this "th" sound like a hard /t/ as in "thing", "something", "think", "thought" (this one sounds exactly like /tot/), etc. in fast speaking situations. They sound like they do place the tip of their tongues between their teeth but no air was blown out of the mouth, thus making the sound of /θ/ somewhat voiced.

So my question is, is this a correct way to produce the sound of the voiceless "th" or is this just a matter of dialect?

Thank you!