It depends a lot on the type and variety of roses you have. Hybrid Tea roses for the most part tend to have that tall and narrow growth habit. It's just the way they like to grow to produce those beautiful long stems for cutting. Floribunda, polyantha and shrub roses have a more naturally bushy growth habit on their own. They bloom in clusters so the plants just grow more bushy. You can try to prune them lower to encourage more branching but in the end the rose will decide for itself how it wants to grow.

It depends on the type of roses you are growing :-) Some by nature will have a tall, slender habit, others will naturally be more bushy. I find with hybrid teas that reducing the height each season to around 24-36" encourages more basal stemming, keeping the plant full and bushy. I have also done severe pruning on ramblers and climbers from time to time to decrease height and also encourage stemming. And fertilizing with alfalfa or a fert that includes alfalfa meal will also encourage basal stemming.

In general, the more robust stems/canes that emerge from the root crown (or above the graft with grafted varieties) the more wide and bushy the plant will be.

What kind or type are they?

Cinco de Mayo (floribunda)

Ch-Ching! (grandiflora)

Red Gold (floribunda)

Should I cut out the older (brown) stems and prune the newer (green) stems?

Cut them down to about a foot at end of winter each year. This will of course make them immediately shorter and will also cause them to have more branches each time they grow back. It won't, however make them depart radically from the comparatively narrow, vase-like shapes that are characteristic of roses in these classes.

Also be sure adequate nutrient levels and other favorable soil conditions are being maintained. If you have been fertilizing with suitable products but not mulching you might want to switch to mostly mulching for awhile, see what response you get to that.

And if there is a rose display garden nearby, where the plants look to be growing in the desired manner you might want to ask workers there what methods are being used.

I apply either Milorganite or a "general purpose" (balanced) fertilizer.