I have come across several Turkish transitive-intransitive verb pairs on DL, but it seems that it is not usually like 'one verb being used transitively, the other verb being used intransitively'. Based on your and Ektoraskan's comments, for example, başlamak may be used both transitively and intransitively, and besides there is another verb, başlatmak, which is used transitively. Are there different forms of transitive grammar which require the use of one verb (e.g., başlamak) versus the other (e.g., başlatmak)?
You could say "I started it six times, but the meaning would be different ("Ona altı kez başladım", I suppose). Then you are talking about all the six times you started it, and you use plural.
"I started it for the sixth time" (= this was the sixth time I had ever started it) or "I started it the sixth time" (= I tried five times and then managed to start it on the sixth try) consider only the single, sixth time, and you use singular.
"for" strikes me as not strictly needed here. Yes, "for the [nth] time" is idiomatic, but so is "Jane started it the first time," "Bob ... the second time," and "I started it the third time." To my ears, including "for" implies that it's the same person doing the starting each time, something that the Turkish doesn't obviously require.