Though except for possible weird stuff with playing cards while it isn't your turn, you're never in a position to choose between starting your extra turn or your normal turn anyway. Extra turns are always generated to happen after the current turn, which is already yours; at the same time your next regular turn would start.
Even then, I don't think you'd be in the position to choose your regular turn. Even if you played an Outpost or Voyage on your opponent's turn (which [I believe] is possible with a Reaction [like Caravan Guard], played using Way of the Mouse, which plays Vassal, which then discards one of them and plays it) in a 3+ player game (so the previous turn wasn't yours), you would still have take your bonus turn(s) first, since they say "...take an extra turn
after this one..." You'd still only ever be choosing from among bonus turns that all come after a certain regular turn.
Although, if you played Outpost that way, you would already have your 5 card hand, with which you would play your extra Outpost turn, and the "next hand" for which "you only draw 3 cards" would technically be for your regular turn (I believe).
There's some misunderstandings of my question, probably because I wrote wrong at first. The question has nothing to do with your next regular turn. I have since corrected what I wrote. Here it is again:
Another Lich question: Do you also skip turns that aren't set up yet when you play Lich? So if I play Outpost, then Lich and then Voyage, can I choose to skip the Voyage turn instead of the Outpost turn?
This assumes that you can choose which extra turn to skip, which I think must be right since you can choose which extra turn to do next.
I think that's right (technically, I think you choose which turn to take next, and then that turn gets skipped by Lich). In my previous post I analogized it to Merchant or Scheme, which are action cards that do something later than when they are finished resolving (in other words, you can play the card, do other things, then get the impact of the card's effect).
You could think of them as creating what I called effect tokens. Each play of Merchant creates an effect token which, if you play a Silver for the first time later that turn, generate +$1. Each play of Scheme creates an effect token which, when you discard a card from play, allows you to topdeck it.*
I think the most reasonable understanding of Lich is that it creates an effect token which, the next time you would take a turn, you instead skip that turn.
* I think this means that if you set aside a Scheme with Prince, you could play Highwayman every turn, by playing triggering Prince first (thus playing Scheme), then triggering Highwayman, using Scheme to topdeck it, and then draw it as one of your 3 cards.