Hi @mks123,
A warm welcome to the Bunnings Workshop community and thank you for your question.
I'm fairly confident that your lawn is predominately Kikuyu grass. What you are seeing is an above-ground Kikuyu runner known as a stolon. Kikuyu grass sends them out in search of uninhabited soil so it can spread its roots and grow laterally. Kikuyu is a fantastic grass type for lawns, as it is fast spreading, hardy and drought tolerant, but it can be quite invasive if not managed.
Unfortunately, there is no way to completely eradicate them without removing your entire lawn and starting again with a different grass type. The best way to deal with them when they pop up where you don't want them such as in pathways and garden beds is to manually remove them, roots and all. If they are within your lawn area, they are working to thicken your lawn so they should not be killed, but regular mowing will keep them in check.
You might like to check out How To Care For Kikuyu Grass for some further information.
Let me know if you have any further questions.
Jacob
