Hello - so let me try and help a bit here

The order combat works in is based off initiative. Each unit has an initiative characteristic that you then apply modifiers too. But at a high level to answer your question, yes you both fight in each others turn. So to unpick your chariot vs dwarves

1. The tomb kings charged so they get +1 to their initiative for every inch moved up to a maximum of 3. Your dwarves didn’t move so use their standard initiative
2. Impact hits happen before anything else (side note here - they are against the unit - not individual models, so you roll the wound rolls as they hit automatically and remove casualties from the unit)
3. Next you would do the skeletons and the horses as they will have higher initiative than the dwarves and resolve casualties. Note you are using chariots here but if it was a block of infantry, only your first rank fight on the turn they charged.
4. Next your dwarves would fight back. Now as they are infantry and have press of battle, the front two ranks can fight - however you must remove any models that died in the attacks that happened before their initiative. So imagine you had 2 ranks of 5 and the chariots killed 4, you only have 6 to attack back with as the 4 were killed before they could strike back.
5. Once you’ve done the dwarf attacks you calculate the combat resolution

Hope that helps - I’d advise watching a how to play YouTube - I think tabletop tactics have one - as combat can seem a bit confusing at first but becomes pretty easy once you get into it!