I want to try to build a multiplayer turn-based browser game.
What technologies do you recommend me to use for server side considering I plan to host it on a shared host? Would PHP be fast enough?
Unless you expect to have millions of concurrent players, I don't think you have to worry about inherent performance differences of languages or technologies. That is even more true since your game is turn based, meaning you don't need real-time interactions, and it's not that bad if something takes a few milliseconds more here or there. If anything, the shared host environment would be a bigger source of problems than the tech you use, because you can't predict when and if your performance is affected.
The best advice is probably: use a language that you're comfortable with.
Also, you can find ready made turn based strategy frameworks such as this one: https://www.assetstore.unity3d.com/en/#!/content/50282