Can I build several connected roads on the same turn?
As stated in the Almanac under "Building":


  You can build as many items and buy as many cards as you desire - as long as you have enough resources to "pay" for them and they are still available in the supply.


So as long as you have enough resources in hand, and roads available to you, you can build out as far as you like in one turn, and then add a settlement wherever you want to place it (while obeying the Distance Rule, of course).

I'd note that while the Turn Overview says for building a road, "place next to one of your settlements or cities", that cannot be correct since then you would never be able to build a second settlement (since you could never build a second road out to satisfy the Distance Rule).