LED lamps draw much less current than incandescent bulbs. This is mostly a good thing, but there are 2 downsides. The original flasher unit blinks according to current draw, so it needs to be replaced. The pilot light on the instrument panel is now a significant current draw, and so the unselected side lamps get a ground connection through the pilot light.
To get you flashers working properly you need to:
1. Change the flasher unit from a current driven type to an electronic flasher. That way you get a steady flasher rate.
2. Either fit a diode on either side of the pilot lamp, or fit a pair of green LED pilot lamps back to back in place of the incandescent bulb.