We have a Lavender plant in a pot in our garden, which is not very
attractive and rather untidy looking. Neither my wife or I are
gardening experts, and therefore do not know what to do to improve its
looks.

It is about 60 to 70cm tall, but the bottom half is basically wood
stem, with dead(ish) looking leaves. From that there are about a dozen
or so tall shoots growing straight up, with greener leaves in groups
of 3 or 4, every 5 to 10 cm. The top 10cm of each shoot currently has
mauve flowers.

Is there a way of promoting growth further down the bush, and what is
the best way to look after this thing?

Cheers and thanks in advance for any help.