How to pronounce "TH" quickly?

First, here are some general things about 'th' (a voiced or unvoiced dental fricative).

• Among all languages of the world, 'th-' is relatively rare at between 4 and 7 percent (depending on sources). Famous languages with 'th-' are English, Spanish (European), Modern Greek, and Arabic (MSA). In comparison it's vanishingly rare to not have a bilabial or velar stop (p/b or k/g).
• By number of individual speakers, however, it is much more common making it pretty familiar to people in general over the world. This is made more complex by 2nd language learners and dialects without 'th-'. The more populous Latin America say 's' where European Spanish speakers say 'th'. And Arabic is a bit of mish mash between a high register Modern Standard Arabic with 'th' but most Arabic speakers speak day-to-day dialects without 'th-'. English is ~500M L1 (th-using) speakers, Spain ~50M, Arabic ~300M... let;s just say under ~1B which gives at least ~
• In historical linguistics, 'th-' tends to be a transitional sound, that is, over centuries the intermediate variety may have a 'th-' but earlier and later ones may not. A common phenomenon is 'lenition' or softening (Latin pater, Italian padre, Spanish padre (with fricative instead of stop), and French père (where the 'd' is softened to nothing).
• In language learning, 'th-' production tends to be one of the last things acquired, children usually using the stopped version ('d' or 't').

All that said, as for English, you're concerned with how 'th-' is pronounced in rapid or informal speech by native speakers. As this situation is not particularly well documented I can only give you my impression as a native speaker (AmE).

The sound is produced as a dental fricative in rapid speech just as in slower inarticulate speech. It is not converted into a stop. It may be dropped altogether.

The most common places that might give you difficulty in rapid speech are in unstressed syllables, or word-initial or word-final contexts where there might be a difficult consonant cluster.

Some examples:

• 'worn thin' - /wɔrn θin/ - even fast, the 'th' is pronounced
• 'in the news' - comes out as /nðnu:z/ - the vowel is elided.
• 'bath time' - the fricative slips right into the stop.
• 'sixth sense'- it is almost impossible for even native speakers to articulate the 'th' in even the most careful and trained speech.

If you are a non-native speaker it is tough enough to get the 'th-' sound when trying hard, but even more difficult in rapid speech. But native speakers don't change it at all.

As with any non-native sound for anybody learning any language in the world there's an annoyingly simple but unsatisfying method for doing sounds outside of your native repertoire - it is to practice a lot. Listen a lot and practice a lot.

Also exaggerate. Over-exaggerate. Exaggerate a little bit more. Listen to some speaker you want to imitate, and imitate them. It may sound like you're making fun of their accent, but really, you probably need to exaggerate. Can't get the trill of Spanish 'r'? Make fun of it. I mean, not to insult but to make a game out of it.

Anyway, even when speaking fast, don't drop the 'th-' or make it into a 'd-'. Native speakers maintain the 'th-'.