This happens because your chrome driver version is different from the chrome version that you are using, Mathematica 13.3 has default 102 and most likely you are using using latest version of chrome (119). I had faced the same issue and this is what I did to get around this,

Start with checking the version of chrome, you can do that by clicking the three dots in chrome -> Help -> About. If your chrome version is above 114 I suggest you to uninstall chrome altogether.

Now you need to get older chrome for your PC and you can find that here (103.0.5060.53 worked for me, I suggest you to download the same as this version should match with chrome driver version, which is to be downloaded next) and install it.

Next you will need chrome driver which is same version as your chrome browser .You can download chrome driver from here.( find 103.0.5060.53 and download it) Find chromedriver.exe in the zip folder downloaded, copy this exe file.

Now find the default chrome driver that mathematica uses for me the path was,

C:\Program Files\Wolfram Research\Mathematica\13.3\SystemFiles\Components\WebUnit\Resources\DriverBinaries\ChromeDriver\Windows-x86-64\102\

Inside this 102 folder there is chromedriver.exe already present, rename this to old-chromedriver.exe and paste the copied chromedriver.exe here. Now chage the folder name from 102 to 103 since we have chrome driver version (103.0.5060.53).

Once everything is done re-launch mathematica and it should be good to use StartWebSession ["Chrome"]