Is the 'th' sound usually reduced in spoken English?

Solution 1:

In actual speech situations, with native English speakers, Fast Speech Rules are unavoidable.

These rules describe normal pronunciations. There are no spaces between words in language,
only in writing, which is not language and does not have anything to do with how we speak.
(After all, most people in the world are illiterate, but they still speak.)

To take just one example, the fraction "5/6" is written five-sixths, and we are taught it's
sposta be pronounced /fayvsɪksθs/.
That's a pretty chewy consonant cluster there at the end:

• /ksθs/ -- a /k/, followed by an /s/, a /θ/, and another /s/. with no vowels at all.
  Phonologically, all are voiceless sounds, and the last three are fricatives.

This makes them hard to pronounce together fast, because the tongue has to go
from the position for the /s/ (touching the top sides of the mouth)
to the position for the /θ/ (touching the bottom of the upper incisor teeth)
and back again to the position for /s/. Very fast, in a cluster.

Such clusters are hard to pronounce and normally get simplified, usually by deletion.
In practice, what people actually do is drop the /v/, and the /θ/, and just say a long /s:/, i.e,

• /fay'sɪkss/

Pious instructions to "enunciate more clearly" do nothing to improve comprehension.
If you are interacting with native speakers, pay careful attention to the way they speak.
They're the ones you want to communicate with.

Solution 2:

It's not a general rule about th sounds. It depends on the actual word.

If it's the first two letters of them, you can reduce it, except in fairly formal speech.

Run 'em all together.

Similarly, you can reduce the first letter of her and him. But this doesn't apply to all th sounds:

The house 'at Jack built,
Jill is smarter 'n Jack,

both sound much more informal to me than dropping the th in them, although I expect both these reductions are common in informal speech. And

somewhere in 'air.

for "somewhere in there" just sounds wrong to me.

These are essentially weak forms of the words. Lots of structural words in English have weak forms; for example, there are weak forms of that, than, but, and just where the vowel changes to a schwa. These weak forms are perfectly acceptable in formal English. (And I believe these are often taught in ESL classes.)

The weak forms created by changing them, him, and her to /əm/, /əm/, and /ər/ are not quite as formal, but I would think they are acceptable in all but the most formal situations. The weak forms created by changing that and than to /ət/ and /ən/ are definitely informal, but still widely used in informal speech. And there is no weak form of there that is missing 'th'.