Here are some recommendations:
SourceTree: https://www.sourcetreeapp.com/
Windows and Mac. Free. Feels sluggish, but is also really dependable, the graph view is lovely and it covers most of the common things that you want to do - also, staging/discarding chunks or even individual lines of code is lovely. Oh, and the Git LFS integration, and creating patches is also really easy. And it gives you the underlying Git commands it uses, in case you care about that.
GitKraken: https://www.gitkraken.com/
Windows, Mac and Linux. May need commercial license. Feels like a step up from SourceTree, but i find that using this for commercial needs is a no go. If that's not an issue, however, it has a good UI, is nice to work with and just generally doesn't have anything i'd object to. IIRC it saved my hide years back by letting me do a ctrl+z for a repo after accidentally forcing to the wrong remote, so that i could fix what i had done (memory might fail me, was years ago), just generally feels intuitive like that.
Git Cola: https://git-cola.github.io/
Windows, Mac and Linux. Free and open source. Perhaps one of the more basic interfaces, but as far as free software goes, it does what it sets out to do, and does it well. I use this on Linux, whenever i want to have that visual feedback about the state of the repo/staging area or just don't feel like using the CLI.
TortoiseGit: https://tortoisegit.org/
Windows only. Free. Recommending this just because you mentioned TortoiseSVN. If you just want a similar workflow, this is perhaps your best option. Honestly, there is definitely some merit to having a nice file system integration, i rather enjoyed that with SVN.
Whatever your IDE has built in: look at your IDE
On any platform that your IDE runs on. Same licensing as your IDE. Some people just shop around for an IDE that they enjoy and then just use whatever VCS workflows that they provide. I'd say that VS Code with some plugins is really nice, though others swear by JetBrains' IDEs, whereas others are fine with even just NetBeans or Eclipse (Java example, you can replace that with Visual Studio or whatever). If you're working within a particular stack/IDE, that's not too bad of an idea.
The CLI: https://git-scm.com/
Windows, Mac and Linux. Free and open source. You'll probably want to know a bit of the CLI anyways, just in case. Personally, i'm still way too used to using a GUI since dealing with branches and change sets just feels like something that's more easy when visual, but the CLI has occasionally helped me out nonetheless.
Actually, here's a useful list of some of the Git GUIs: https://git-scm.com/downloads/guis
For example, did you know that there are some simple GUI tools already built in, git-gui and gitk?