You can install any Sinhala font from your mobile itself and use it in any browser you like with following instructions.
1) Root your device.
I rooted my Sony Xperia V (LT25i) with "towelroot" tool. No need to run complex scripts, just one click is enough.
2) Download a Sinhala ttf font to your phone and rename it to "DroidSansFallback.ttf" (or get a renamed file from here)
I have put this font file to "/sdcard1/DroidSansFallback.ttf" in the beginning.
3) Install a terminal emulator app from Google play store", I used this "terminal emulator"
Open the terminal emulator app and run the following commands in the given order.
1) Get root access
2) Grant super user access to terminal emulator in the pop up window.
3) Get read/write access to /system
#mount -o rw,remount /system
4) Copy DroidSansFallback.ttf file /system/fonts
#cp /sdcard1/DroidSansFallback.ttf /system/fonts
Now you can open your favorite browser and read Sinhala text, I used Firefox browser to test it.