The lag is due to the computer continually accessing the hard drive while you are working.
If you ran a recovery after formatting odds are good that recovery installed a lot of useless stuff, so you might want to consider uninstalling all the tools, helpers, reminders and other nonsense running in your taskbar that you don't actually need.
Another thing to try is disabling hibernation mode, hibernation mode is constantly making backups of whatever you are doing so that you can close the laptop and resume without having to reload everything, but as a result it is constantly accessing the hard drive which can adversely affect performance. In the start search window type cmd, in the list that comes up right click on cmd.exe and select Run as Administrator, this will open a command prompt window, in this window type "powercfg /h off" without the quotes. Restart your computer and see if there is any improvement.
When I was doing tech support for HP one the first things we would have customers do is a Hard Reset, disconnect the power and remove the battery then push and hold the power button for 20 seconds to completely discharge the circuitry as well as release any buildup of static electricity.
Then we would have the customers open the access panels on the bottom of the notebook to re-seat the hard drive and ram, by removing these components and putting them back in, this would clean any oxidation of the associated contacts that could be causing intermittent problems.
Then it's time to put it all back together and see how it works.
Also make sure that the notebook has adequate ventilation, laptop is a misnomer since putting the device in your lap will block ventilation ports causing the device to overheat, overheating can seriously degrade performance or even result in component failure.
If there are still problems the first thing you want to do is run a full scan with an anti-virus program, viruses can cause lag spikes of this nature. But in this case there is a distinct possibility that you have a defective hard drive.
Boot to the BIOS and run a hard drive diagnostic, the diagnostic usually consists of 2 parts, the first part is a short diagnostic, this can optionally be followed by the long diagnostic. The short diagnostic only takes a few minutes and if there is a serious problem with the hard drive it will stop and give an error fault code, if there is a fault code at this stage then as long as you can access the computer I would recommend backing up your data since the drive will need to be replaced. If however the drive passes the short test, then I would recommend running the long test, this test can take hours so don't start it unless you can leave the computer running, like say over night. In either case make sure you write down the error codes if any. If the laptop is under warranty the manufacturer will require that you run these tests anyway, so it will save time if you do this in advance.
If the laptop is no longer under warranty then a Solid State Drive is a good choice as long as it's understood that SSDs are more expensive than their mechanical counterparts, but they are definitely worth the investment.