GE Powermark Electrical Panel - What Did the Last Guy Do?
Ground bar fun and games

The original installer's decision to fit a separate ground bar for the ground wires was a good one from a neatness standpoint, even though Code does not take issue with having all the grounds and neutrals on one bar at the main panel, as that's the only place where neutral and ground connect (via that bonding strap you see bottom left).

However, the choice of ground bars and ground bar locations available for your panel leaves us in a pickle.  None of the ground bars your panel accepts provide more than 14 connection points, and your panel is not labeled for multiple ground wires in a single ground bar hole as per NEC 110.14(A):


  Terminals for more than one conductor and terminals used
  to connect aluminum shall be so identified.


The only way I can explain this is that your panel predates that marking requirement in Code, leaving us with three not-great options if a subpanel is off the table given that your panel also does not have a factory provision for a second ground bar:


Stuff the new ground wires into unused holes in the neutral bar (one large, two small) in an annoyingly sloppy fashion.  This is the simplest option, and also the easiest to make Code-compliant right now, but has the downside that it'd make life harder for anyone who wants to convert this panel to a subpanel.
Rearrange the wires in the existing ground bar to free up one large and one small hole. This requires a bit of care to get the torquing right, since you are operating outside the loadcenter's labeling at this point, and also is technically a violation of the loadcenter's listing, although not any more so than the current situation.
Fit another ground bar to the panel (in-line with the existing bar is fine) and land your wires there, despite the fact your panel doesn't have provisions for this and isn't labeled for multiple ground bars.  A GE TGL2 kit can be used for this, although you will have to switch the machine screws used to mount it out for a pair of 10-32 (i.e. fine thread) self-tapping/self-drilling screws, such as Garvin GSSTs.  Note that ordinary (coarse thread) sheet metal screws are not suitable for this, as they will not engage two threads into the sheet metal of the cabinet as required by NEC 250.8(A) point 6:



  (6) Thread-forming machine screws that engage not less than
  two threads in the enclosure


Given these alternatives, I would choose the third if possible and the second if not, but this is a case where your local Code inspectors probably can provide more guidance than I can.

Other option: run a feeder to the garage

The other thing I would consider at this point is running a feeder from this panel to a subpanel in the garage.  This has the advantage that you will only need one grounding hole free in the main panel instead of a minimum of two, at least, and it also keeps the space usage in this already fairly tightly packed main panel to a minimum.

I would go with a 125A feeder here (using 1/0-1/0-1/0-1 aluminum SER cable and a TGL20 lug in the spare ground hole you free up for this, as well as a 286A8894G1 add-on neutral lug attached to the bar for the top row of neutrals) to a 125A, 30 space panel, in order to provide adequate room for expansion.  (Going beyond 125A is rather difficult here due to GE TQDL breakers being hard-to-find, and your panel cannot accept a 150 or 200A subfeed lug block it appears, for that matter, nor will it accept a 200A TQDL breaker.)