There is a new expressway/tollroad under construction currently creating a new western beltway around greater Jacksonville, through Clay County near Green Cove Springs crossing a new bridge over the St Johns River and ending with a major interchange with I-95 in St Johns County.
Details on the construction status, routing and planning can be found here.
http://firstcoastexpressway.com/At the moment it is signed as FL23. The tollway from the Cecil Commerce Center to Blanding Boulevard is just more than 50% complete.
The current ROW end at Blanding Boulevard is overpass complete with ramps still under work.

The ROW going north of Blanding is nearing completion with the base pavement down and installation of the ePass overhead grids in place.

FL23 gets bounced back and forth depending on the completion level of the ROW.

As far as the expressway south of Blanding to Clay County and over to I-95, it will require a new bridge over the St John's River to replace the current 2 lane (narrow) Shands Bridge.

This is still in planning and there is still much ROW to acquire in Clay County. There is also some governance issues on the fact a free bridge (Shands) is being replaced by a Toll Bridge. People on the east bank of the St Johns River cross the bridge to go to Green Cover Springs for retail frequently. Lack of retail in ex-urban St Johns County is an issue currently, but many believe will be resolved when I-795/CR2209 is completed.
The schedule for the rest of the expressway is as follows:
First Coast Expressway (I-95 to Blanding Blvd, Currently Being Designed)Construction Cost (estimated): $897 Million
Schedule:
Right of Way acquisition (estimated completion by 2020)
Design and Permitting (estimated completion by 2020)
Construction (not currently scheduled)
Project Improvements:
Construct a multi-lane divided expressway from I-95 to Blanding Blvd.
Construct a new crossing of the St. Johns River south of the existing Shands Bridge.
Construct drainage ponds and an underground storm water system.
Build automated tolling facilities for electronically paid tolls.
There are discussions on if this expressway will be extended north into Nassau County and reconnect with I-95 somewhere near Yulee.
Naming conventions are currently under speculation. It will stay signed FL23 indefinitely. If the road stays a I-10 to I-95 beltway, current thoughts are it will signed I-310.
If they are able to complete the beltway all the way into Nassau County some think it will be a derivative of I-95 instead of I-10.