I've heard this phrase a few times, "(kamlang) tawd saai-dtaa". Literal is frying one's eyesight(?)

From various dictionaries:

THAWT SAI TA ทอดสายตา - To cast one's eyes (on, over); cast a look ; look at.

NB

THAWT ทอด can also mean to cast.

From various dictionaries:

THAWT SAI TA ทอดสายตา - To cast one's eyes (on, over); cast a look ; look at.

NB

THAWT ทอด can also mean to cast.

Ta Tua ตาถั่ว - You don't look / see things properly

I think we would use ทอดสายตา for when we're gazing into the distance, ie a singer on stage looking out over the audience.

In contrast there's ชำเลือง to look at sideways, ie cheating in an exam by looking at the next person's paper and เหลือบ(มอง) to glance at quickly, ie a clock when you enter a room.