In SimCity, there are two different types of road. These are streets and avenues.
Streets are one square wide and avenues are two squares wide. All streets are the same width, so a dirt road can be upgraded to a high density street using the upgrade road tool.
In order to upgrade a street to an avenue, however, you need to destroy the old street and replace it with a larger avenue, which will remove all infrastructure that exists along the old street.
If you're in a situation where you already have the maximum capacity version of a particular style of road but don't have the space to replace it with an avenue, there are other options available. It is possible to build additional roads within the vicinity to alleviate the amount of traffic travelling down your currently over-crowded road. To do this, bring up the traffic data layer to see where all of the traffic is coming from, and build new roads that intersect the busiest streets to provide additional routes for your sims to travel down. The traffic data layer can be accessed using the 'All Data Maps' button in the bottom right of the interface, or by toggling the 'upgrade road' option.
In addition, there is the possibility of setting up mass transit. Mass transit works by building a depot (and adding additional trucks as required to increase the number of sims that can travel per day and reduce the wait time for busses) and placing stops. The region that a stop will cover is highlighted along the roads as you place the stops. For the maximum effect you will want to place stops along your busiest streets (see your data layer for traffic again) and ensure that the wait time for passengers is managed to ensure it doesn't get too high.
You can monitor wait time by bringing up the information screen for your mass transit depot, and you can monitor the amount of traffic a particular stop is attracting by bringing up the information screen for each individual stop.