My link strength is 1. I run into this ice. The Corp pays 1 credit to raise the trace to 5. I pay 4 credits to raise my link strength to 5. I pass the first subroutine. At this point does my link go back to 1 so we would have to start paying credits again?

They are separate traces. Each follows the procedure on p. 20 of the core set rulebook independently.

OK, so basically each trace subroutine starts the process all over again.

Not trying to be flippant or glib here, but is there anything in the rules about subroutines or traces that would indicate otherwise? Reread the rulebook and think about how to apply those processes in the situation you describe.