Very expensive roaming mobile internet on your home sim card should work if you want to access the internet in Turkmenistan. It is also possible, if not easy, to buy a Turkmen sim card.
Your updates and questions regarding phones and internet in Turkmenistan are welcome in this forum topic. The best resource for sim card knowledge is the prepaid data sim card wiki.
The latest we heard (2019) was that a Turkmen sim card costs 10 manat for locals and 30 dollars for foreigners. It is possible to pay a local to get a sim card for you in their name, if you so desire. Speed is acceptable.
Internet and wifi
Internet in Turkmenistan is filtered. You are being watched and unwanted websites are blocked (Caravanistan is not blocked anywhere in Central Asia or China by the way). Most social networks are also blocked, including Whatsapp and Telegram (Turkmens use Imo). VPNs are blocked, we haven’t heard of one that works.
Generally, there’s no wifi in hotels or restaurants. There seems to be 1 internet cafe per city.
There are a few good places for wifi in Ashgabat. Fast free wifi is available at the coffee bar in the Sofitel Ashgabat Oguzkent Hotel, or get a coffee at the Grand Turkmen Hotel. Even better is the Public Affairs section of the US Embassy (3rd floor, Business Center, Andalyp street, 70) Open Mo-Fri 9-13 and 14-20, Sat 09-12 and 13-18.
Everywhere else, it will be highly frustrating. A few high-end hotels in Ashgabat, Balkanabat, Mary and Turkmenbashi have internet, but it’s very expensive (5$/hour is normal), very slow and unreliable.