Here's the LTspice simulation of a 555 circuit that I think does what you want.
The circuit uses one 555 timer, and two CD4027 JK flip-flop chips, plus resistors and capacitors.
Outputs B1, B2, and B3 control the bulbs (B1 is inside).
It continues the sequence as long as the Turn signal voltage is high (simulated by V_Turn source on left).
The values of R2 and C5 determine the speed of the sequence.
The values shown give a complete sequence time of about 0.9 seconds.
Higher values slow it down.
You will need to add a transistor driver to each output for whatever the bulb current is.
(If you know the bulb current, we can help with that. Are they incandescent or LED?)
Power and ground for the CD4027 chips are not shown but must be connected.
Be sure and tie all unused inputs on the spare 4027 to ground to avoid any erratic behavior.
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