Seems there is a reed in the speedo that sends a signal to the tsm as to how far you have gone since you actuated the turn signal this is a white / green wire that comes from the speedo and goes to the tsm it is pin no 2 on the speedo connector block and pin no 3 at the tsm connector
so there should be at least 2 wires from the speedo as it seems to share a common ground with the tacho on the 883 there are 3 wires coming from the speedo the white with a green trace is the one that is perhaps faulty or could be the reed not sure from this distance
support the bike with the front wheel off the ground connect an ohmmeter between the speedo white/green wire and ground spin the front wheel watch the scale it should alternate between 0 ohms and infinity if that is ok then check that there is 12 volts to the tsm end of the connection by unplugging the tsm connector and testing between pin 3 white/green and ground if there is 12 volts then the tsm is faulty if you do not have 12 volts then there is a wiring fault
I hope this helps a wee bit, be aware i only know a wee bit about this electrickery stuff
Brian