Hindu scriptures prescribe the joint family lifestyle.
To my knowledge there are no explicit verses in the Smritis which say that one must live jointly but several verses can indirectly be interpreted leading to that conclusion.
Manu Smriti 2.227. That trouble (and pain) which the parents undergo
on the birth of (their) children, cannot be compensated even in a
hundred years.
Manu Smriti 2.235. As long as those three live, so long let him not
(independently) perform any other (meritorious acts); let him always
serve them, rejoicing (to do what is) agreeable and beneficial (to
them).
Manu Smriti 2.236. He shall inform them of everything that with their
consent he may perform in thought, word, or deed for the sake of the
next world.
These above verses are hinting towards all sons staying with parents lifelong.
Manu Smriti 3.116. After the Brahmanas, the kinsmen, and the servants
have dined, the householder and his wife may afterwards eat what
remains
3.116 is saying that the husband and wife should dine only after the relatives (Svayeshu in the original text) have already dined. This is clearly talking about a joint family.
Manu Smriti 2.132. (The feet of the) wife of one’s brother, if she
be of the same caste (varna), must be clasped every day; but (the
feet of) wives of (other) paternal and maternal relatives need only be
embraced on one’s return from a journey
Now, clasping the feet of the wife of one's brother is possible only if the families are staying together.
The Vedas too clearly advocate the joint family system:
In the Atharva Veda (denoted by AV in this answer), there is Concord Hymn, which talks about maintaining unity and concord in a joint family.
The union of hearts and minds, and freedom from hate i'll bring you.
Love one another as the cow loves the calf that she has borne. AV
3.30.1
Let not a brother hate a brother, nor a sister hate a sister;
unanimous, united in aims, speak you words with friendliness. AV
3.30.3
Having superiors who are conscious (of their tasks), be not divided.
Thriving together, moving on with joint work, come speaking sweetly to
one another; I'll make you have one aim and be of one mind. AV
3.30.5
I will make the prayer, for that concord among men at home, by
which devAs are not apart from them, so that none hates the other.
AV 3.30.4
Common be your water-store, common your share of food. I bind you together to a common yoke. United, gather round Agni, the spokes
around the nave of a wheel. AV 3.30.6
The bride's role in a joint family is explicitly stated in the following mantra:
samrAjnedhi svasurishu (1),
SamrAjnuta devrushu (2),
NanAndu
samrAjnedhi (3),
SamrAjnuta svasravAha (4).
............
Be a queen to thy father-in-law (1), a queen to thy brothers-in-law
(2), a queen to thy husband's sisters (3), a queen to thy
mother-in-law (4).
AV 14.1.44
Note that SamrAjni= Queen. So, the mantra is asking the bride to be like a Queen (means very dear) to her in-laws.