Demand, Recommend, Insist and Suggest: Combinações possíveis

Hello guys

Lately, I've seen a site on the internet saying that the verbs "To demand", "to suggest", "to insist" and "to recommend" had a different pattern. The combination is:

noun/pronoun + Verb to demand/insist/... (+ that) + noun (subject pronoun) + verb in infinitive form without to

Examples:

I demand (that) she take a painkiller.

His friends insist (that) he call her.


I thought it'd sound strange if I used the subject pronouns "He", "She" and "It" without adding an "S" to the verbs performed by them in the present.

Does this combination really exist?