
dadurch
thereby
Introduces a result or consequence of something mentioned before; therefore, as a result.
Er hat hart trainiert und dadurch den Wettkampf gewonnen.
He trained hard and thereby won the competition.
thereby / through it
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thereby
Introduces a result or consequence of something mentioned before; therefore, as a result.
Er hat hart trainiert und dadurch den Wettkampf gewonnen.
He trained hard and thereby won the competition.

in this way
Describes the means or method by which something is achieved; in this way.
Man öffnet die Flasche dadurch, dass man den Deckel gegen den Uhrzeigersinn dreht.
You open the bottle by turning the cap counterclockwise.

through it
Describes movement through a specific place or opening.
Das Tor war offen, und wir konnten einfach dadurch gehen.
The gate was open, and we could simply go through it.
'Dadurch' is made up of 'da' and 'durch'. The literal meaning 'through that' helps you understand the figurative meanings too: a consequence arises 'through a cause', or a result is achieved 'through a method'.
Der Eingang war schmal, aber wir passten dadurch.
The entrance was narrow, but we fit through it.
As an adverb, 'dadurch' can stand at the beginning of a sentence (position 1). In this case, the verb follows directly in position 2 (inversion).
Die Straße war gesperrt. Dadurch mussten wir einen Umweg fahren.
The road was closed. As a result, we had to take a detour.
To describe a method, 'dadurch' is very often combined with a subordinate clause that begins with 'dass' (that) or 'indem' (by). 'Dadurch' then introduces, in the main clause, the result of the action described in the subordinate clause.
Er verbesserte sein Deutsch dadurch, dass er jeden Tag deutsche Nachrichten schaute.
He improved his German by watching German news every day.
The spatial meaning 'through this place' is always tied to a concrete context. An opening, space, or passage that 'dadurch' refers to must have been mentioned before.
Das Tor stand offen und die Schafe liefen einfach dadurch auf die Weide.
The gate was open, and the sheep simply ran through it into the pasture.
so that / with it
Damit and dadurch can explain how a result is achieved. Damit tends to refer to an object or means being used. Dadurch emphasizes more strongly an approach or process that brings about the result.

Wir schneiden das Gemüse in kleinere Stücke, dadurch wird es schneller gar.
We cut the vegetables into smaller pieces, so they cook faster.
Wir benutzen ein scharfes Messer, damit schneiden wir das Gemüse schneller.
We’re using a sharp knife; with it, we can cut the vegetables faster.