Your blinker should NOT be causing all the stuff in your instrument panel to go off like that. You have a floating voltage seeking ground through whatever is available. Check the grounds on the turn signals first. Also, see if there is a SHORT between a 12V wire and your instrument cluster. Specifically, look at the right turn signal wire. Try disconnecting it at the turn signal switch, then see if the evil behavior of the instrument panel abates when your turn the turn signal on.
I suspect a short here. Bad grounding might contribute. Dave is right about insuring your cluster is WELL grounded. Make a bond wire to ground it through the firewall to the engine block, or even to the battery if you like. Test Dave's hypothesis by rigging a bond jumper with a pair of alligator clips and a good #12 AWG wire going to the negative terminal of the battery, then try the turn signal. I LIKE bonding jumpers, and consequently Mathilda has few wiring troubles attributable to lack of ground.
Replacing the old power supply for the cluster isn't a bad idea either. You can make one with an LM555 op amp and a power transistor, or buy a ready made 12V -> 5 or 6 V supply which will provide some current (7-10 A should do) to the cluster. I got rid of my cluster entirely and will eventually make one with all 12V instruments. I like the LOOK of the old one, but instrumentation is ONE aspect of driving where I prefer modern devices, albeit, simple, rugged ones. Those old thermo-electric gauges left a bit to be desired...... although a working instrument cluster at night is damned pretty.