Hello,

This could be the result of drought damage or disease. You can cut out the dead branches, but I’d go to this page first to see if your plant has other problems:

https://apps.extension.umn.edu/garden/diagnose/plant/deciduous/dogwood/branchesdeadordying.html

If we have another very dry summer make sure you regularly water the dogwood.

Good luck.

You’re welcome!

This is a good guide from the U for a renewal pruning of your dogwood. 

Renewal pruning for older or overgrown shrubs

• Every year remove up to one-third of the oldest, thickest stems or trunks, taking them right down to the ground. This will encourage the growth of new stems from the roots.
• Once there are no longer any thick, overgrown trunks left, switch to standard pruning as needed.

Rejuvenation pruning for older or overgrown shrubs

Deciduous shrubs that have multiple stems (cane-growth habit), and that have become very overgrown or neglected can be rejuvenated by cutting all canes back as close to the ground as possible in early spring.

That season's flowers may be sacrificed but the benefits from bringing the plants back to their normal size and shape outweigh this temporary collateral damage.

This pruning technique works best for shrubs such as overgrown spirea, forsythia, cane-growth viburnums, honeysuckle and any other multiple stemmed shrubs that are otherwise healthy. Within one growing season, these shrubs will look like new plantings, full and natural shaped

Hope this helps.