I think gjr0 might have the answer. There are two versions of the two terminal flasher relay.
One style has a resistive wire in series with the bulbs that heats up a bi metallic spring and causes the relay contacts to close. The contact closing illuminates the bulb with full voltage; but, also passes the resistive heating element which starts cooling down and pops open the contacts turning off the light and sending current through the resistive heater again to repeat the cycle. If you have insufficient initial current to heat the resistor this relay will fail to turn the signal light on initially because the contact never goes closed because of
The second style works similarly; but the relay contacts are initially closed and light up the turn signal bulbs right away. In this relay the resistive heating element is energized and heats up the bi metallic spring when the contacts are closed (bulb illuminated). When the bi metallic spring heats up it bends and opens up the contact shutting off the resistor which allows it to cool and the contacts to close.
If you have the first style relay, insufficient initial bulb current will result in the contacts not closing and the front turn signal should not operate. Since your front turn signal illuminates continuously, I expect that you have the second style relay. However, I expect that you may have two problems. This type of relay should have a form of hyper flash when one bulb fails. Because of the lower current with one burnt out bulb the heating resistor take longer to pop open the contacts and the contacts close back faster resulting in a long 'on' followed by a short 'off'. This hyper flash is actually a design feature and is a regulatory design requirement in North America. It seems like your single front bulb is not drawing enough current to heat the resistor to pop the contact open. This could be caused by
- you front signal light sockets are similarly corroded to the back resulting in a high resistance connection
- you have the wrong (low wattage) bulbs or LED bulbs up front
- your turn signal relay has seen better days.
If your right rear is not illuminating at all and you have correctly replaced the socket, then you have a wiring issue in the circuit to the right rear light. Mr. Ron K has conveniently provided wiring diagrams for the Amazon.
https://www.sw-em.com/Wiring_Diagram...Wiring_Diagram
Since your right front turn signal illuminates, I suggest that you start checking those junction connectors shown in the wiring diagram which connect the right rear to the common junction to the right front which we know works.