All of your images pull from the secure ... is there a reason for that?

Your New Products is 4900+ products ... do you really have 4900+ products or have you never set a proper limit in the Configuration ... Maximum Values ...
New Product Listing - Limited to ...
Limit the New Product Listing to
0= All Products
1= Current Month
7= 7 Days
14= 14 Days
30= 30 Days
60= 60 Days
90= 90 Days
120= 120 Days
On your Index Listing ... do you have the settings for the Featured, New, Specials and Upcoming turned on or off? Off is 0 and On is the values assigned for the display order ... if not using these, set them to 0 for additional speed ...

Have you tried turning on the Configuration ... GZip feature?

While you do not show Category counts, do you actually have them turned off in the Configuration ... My Store ...
Show Category Counts
Count recursively how many products are in each category

true
false