Signal light flashers operate on the principal that when a load is applied to them (the resistance of a bulb) they heat up to open the contact thereby making them flash. But your new LED bulbs draw so little that they can’t heat up the element enough to open the circuit. Or, if it’s a late model vehicle, the built in warning system sees no load and warns you by flashing fast so you know you have a problem. The solution to your problem lies below.
Add a load resistor to each circuit that had a bulb replaced with an LED bulb. Solder one in place between the signal light wire and the ground wire, add a couple of screws to hold them down to a piece of metal and problem solved. *** Warning *** Load resistors get hot. Mount them on something that can take the heat.