I've done many T4/T4S connections on separate tickets at MAD with no issues but I am not facing a T4 to T1. We will be flying from the US to Madrid landing at 9:35am and have a connection on Turkish to Istanbul at 11:50am. We'll be flying business on US-MAD and coach on TK if that matters. All hand luggage.
We're all young and can hustle, are we at the mercy of an on-call bus here?