<PREFACE>
The most widespread solution seems to be placing a modified d3d8.dll file from some random link in the game's folder. Another solution for the bad performance is to use the DxWrapper project, available in https://github.com/elishacloud/dxwrapper.
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Originally got this fix from a user "Tom" on this forum at

https://gtm.steamproxy.vip/app/4760/discussions/0/2632849816277787179/

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I'm just going to copy paste his fix along with my own addon.

"Unfortunately, this game seems to have a lot of lag and performance problems on newer systems. The most widespread solution seems to be placing a modified d3d8.dll file from some random link in the game's folder. Although this solution works, it is never a desirable solution to put some random dll on our system. In this case it is not clear how it works and where does it comes from (I tried searching on Google the author or the original source but no luck).

Another solution for the bad performance is to use the DxWrapper project, available in https://github.com/elishacloud/dxwrapper. This projects allows converting Direct3D 8 to Direct3D 9, which will solve the performance issues of this game. To install it, download DxWrapper.zip from https://github.com/elishacloud/dxwrapper/releases, and put the files dxwrapper.dll, dxwrapper.ini, and d3d8.dll inside the game's folder. Open the file dxwrapper.ini and change the option "D3d8to9" from 0 to 1 to enable the conversion of Direct3D 8 to 9. Now you can launch the game and it should be running fine."

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from user 'playerdav'

If you want a smooth transition from game to other windows: in the d39d section
[d3d9]
AntiAliasing = 0
EnableVSync = 0
EnableWindowMode = 1
FullscreenWindowMode = 1
WindowModeBorder = 1

Save the file. Close it.

Make sure you have the same resolution for the campaign map and the battle map. This can be checked in the preferences.txt file.

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from user "Sebasta"

If your game freezes after playing the intro movies.
You might be missing some DirectX files.

Downloaded them via Microsoft DirectX® End-User Runtime on

https://www.microsoft.com/en-us/download/details.aspx?id=35

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from user "StolenEagle"

if after all this everything run fine but you are missing projectiles and flame projectiles, and getting white blotches on the ground with grass enabled, try downloading a more recent version of the file, which can be found here

https://github.com/crosire/d3d8to9/releases

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Everything should be working fine, but just for posterity, back when I played RTW back in 2016,

I used a dll that I downloaded from a secure site

https://drive.google.com/file/d/1MYRpi2-uPtty6c6brxZKAUi9GbPS3Z4t/view?usp=sharing

This is the file i personally uploaded from my game backup files.

Put this file d3d8.dll in <steam/steamapps/common/Rome Total War Gold> folder.

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Again, all updates are from this thread here:

https://gtm.steamproxy.vip/app/4760/discussions/0/2632849816277787179/