Learn Turkish: Yeniden başla, lütfen.

It is built from the adjective yeni, which means 'new', and the suffix -den ('from'). So yeniden literally means 'from new', which translates perfectly to 'anew', 'again', or 'over' in English.

Since the lesson is about the negative imperative, how would I say 'Don't start over'?

To tell someone not to do something, you add the negative suffix -ma or -me to the verb stem. The stem here is başla. Because its last vowel is 'a' (a back vowel), we add -ma to get başlama. So, 'Don't start over' would be Yeniden başlama.

Why is the verb başla just the bare stem of the word?

In Turkish, the informal singular command (telling one person what to do) is exactly the same as the verb stem. To find it, just take the dictionary form of the verb, başlamak (to start), and remove the -mak infinitive ending. That leaves you with başla (start!).