From State Dept: "Because dual citizenship is not recognized, dual nationals are often denied departure from the country...until they renounce their Turkmen citizenship, a process that can take six months to a year or more." U.S. policy on dual nationality can be found
Embassy notes that, along with a visa application: "A letter of invitation from the person or company that you are visiting is also required...certified by the State Migration Service of Turkmenistan, and it will only be valid for 3 months from the date of certification. If you are only a tourist and will not be visiting anyone in particular, then you need to contact a tourist agency and obtain permission from them." Visas are available for stays up to 12 months.