So, you want to do a large number of curve fits. Can you be surprised that it takes some serious effort? Just wanting something to run faster is nice. As the saying goes however, if wishes were horses, beggars would ride.
If you want to see more throughput, I have shown how to do what I call batch modeling, where you solve multiple problems at once. It is described in my optimization tips and tricks document on the file exchange. However, you need to recognize that to get a speedup here, you will need some skill in the use of sparse matrices, with the optimization toolbox, and with MATLAB in general.
Next, in that same tips and tricks document (as well as my fminspleas tool on the FEX) I show how to use a partitioned least squares scheme to speed up the convergence of nonlinear least squares problems.
You could use tools like the parallel computing toolbox. That too can yield some speedup, IF you have that toolbox, AND you can use it effectively, AND you have sufficient processors to gain speed.
Finally, you can use tools like GPU processors to give some speed, but here too, you will need the proper tools and the knowledge of how to use them.
In all cases, you need to invest some effort to get that speedup. If things were easy, they would already be done for you. You will also need to make sure you have good starting values for each sub-problem, as that can aid in both robustness and in speed of convergence.