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I've seen some sentences, and some of them use 上网:
通过上网聊天，我认识了很多新朋友
Through online chat, I meet many new friends
whereas for another case, it uses 网上:
网上购物是最流行的生活方式
Online shopping is the most popular lifestyle.
In both sentences, they mean as "online". But I'm a bit confused on the word order, when should we use 上网 and when should we use 网上?
上网 is a Verb Object construction internally ("ascend net" = to get on the web). 上网购物 is a Serial Verb construction: first action 上网, second action 购物. Both actions are equal, not subordinate/main, as both are needed. "Serial" because they are chronological (need to 上网 first, before you can 购物, another Verb Object construction internally). A similar construction: 上车买票 "ascend vehicle buy voucher" = get on the bus/tram/train and buy the ticket).
网上 is an Adverb (answers the question "where": "where buy things?" --> 网上购物 "net on buy thing" = to shop online). Can be an Adjective (what kind of shopping? online shopping).
上网 is used when someone is doing the action of going online. 网上 (or sometimes 网路上）means online / on the web.
For instance, of the two examples you provided, you could use 网上 or 上网 under different circumstances.
通过上网聊天，我认识了很多新朋友
You could use "网上" to express a similar meaning:
我在网上认识了很多朋友。
And
网上购物是最流行的生活方式
You could use "上网" to express a similar meaning
很多人喜欢上网买东西。
Hope this helps clear things up! :)
上网 and 网上 are interchangable in the two cases but not all, the meanings are similar but different in Chinese. 上网 literally means go to the internet in Chinglish, while 网上 literally means on the internet. The word 上 in 上 + somewhere structure is a verb like that to go in English, generally means go to or get on, like 上网, 上车, 上楼, 上学, 上课. The word 上 in somewhere + 上 structure is a preposition like that on or upon in English, like 网上, 车上, 楼上, 地上, 桌上.
上网==上+网，here 上 is a verb, meaning surf, go on
网上==(在)+网+上，here 在...上 means on, a preposition. But in most time, 在 is dropped off. In the same way, 桌上==在桌上， 纸上==在纸上,here 上 after nouns is the preposition.
Moreover, 线下==(在)线下，碗里==（在）碗里，屋外==(在)屋外
上网, literally "to get on the internet" [上=(verb)to get on something, 网=(noun)internet], means to surf on the internet.
网上, literally "on the internet" [网=(noun)internet, -上=(postposition)on something], means to be online.
上网聊天 = to use the internet in order to chat.
网上聊天 = to chat online
