It quite simply is not standard English orthography to put multiple utterances by different speakers in the same paragraph.
I know it's not what you want to hear, but we have all had the rule drilled into us since middle school: New speaker -- new paragraph. Every time.
The exception is reportative first person, where a narrator is telling others what a person said or did; in that case the speech counts as the person speaking "now", not the person speaking "originally".
This makes a lot of written English have many, many paragraphs when there is dialogue. That's just the way it is. The only alternatives are to "break the rules", and most English readers will find this very jarring or confusing. Certainly if you just used the Russian emdash most English readers would not know that the speaker had changed - and even when they understood what you were doing, it would still be slow and difficult to read and seem rushed or jumbled.
There are some authors that do this with frequency, precisely to generate that jarring effect or confusion. But they would fail a writing class.