The word RELATIONSHIP does not really have a short form but you can see some people writing the word as Rel. Everybody has his or her special way of writing words which means I can decide to reduce the length of any word to anything as long as I won't mistake it for something else. For example, at times I use Compreh instead of comprehension, also 'compd' instead of compound.
Relationship means a form of association or connection you have with something either blood relationship or social relationship. One special thing about a man is the ability to form a relationship with another person. Some relationships are just casual while some are intimate. I strongly believe whichever way we might choose to abbreviate the word 'relationship', as long as it is making sense and someone else can spot it as relationship, there's no problem abbreviating it.
There are some people who abbreviate relationship as rel. Although this is not an official abbreviation of the word. There are a lot of people who feel that different relationships are important. Relationships are formed in different ways. There are people whom we can build friendships with that will last for a lifetime.
There are also some people whom we will be in a relationship with for a short period of time. A lot of people think of relationships as something that is always related to romantic love but this is not true at all. You can form a relationship with your pet or even with a plant and this will still be okay.
I do not know of any abbreviation for relationship, but you know, written language is there for you to use as best fits you. This is what humans have done over centuries. I personally abbreviate therefore with 3 dots and because as a triangle. I use both a lot.
If you use the word relationship a lot, and it does have many letters, you could put rlsp and for people who might read what you write, add it in full in brackets afterwards the first time you write it. I use cd for could. Writers who pen thousands of words a day have to find these short cuts to make life manageable.